From susceptibility to frailty in social networks: The case of obesity

Abstract

The obesity pandemic is represented by a discrete-time Hopfield Boolean network embedded in continuous-time population dynamics. The influence of the social environment passes through a system of differential equations, whereby obesity spreads by imitation of the most influential neighbors, those who have the highest centrality indices in the network. This property is called “homophily.” Susceptibility and frailty are redefined using network properties. Projections of the spread of obesity are validated on data collected in a French high school.

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Mathematical Population Studies
An International Journal of Mathematical Demography
ISSN: 0889-8480 (Print) 1547-724X (Online) Journal homepage: http://www.tandfonline.com/loi/gmps20
From susceptibility to frailty in social networks:
The case of obesity
Jacques Demongeot, Mariem Jelassi & Carla Taramasco
To cite this article: Jacques Demongeot, Mariem Jelassi & Carla Taramasco (2017) From
susceptibility to frailty in social networks: The case of obesity, Mathematical Population Studies,
24:4, 219-245, DOI: 10.1080/08898480.2017.1348718
To link to this article: https://doi.org/10.1080/08898480.2017.1348718
Published online: 01 Dec 2017.
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From susceptibility to frailty in social networks: The case of
obesity
Jacques Demongeot
a
, Mariem Jelassi
b
, and Carla Taramasco
c
a
Université Joseph Fourier, Grenoble, Ageis, La Tronche, France, and Escuela de ingeniería civil en
informática, Universidad de Valparaíso, Chile;
b
Université Joseph Fourier, Grenoble, Ageis, La Tronche,
France, and Université Tunis el Manar, Institut Pasteur de Tunis, Tunisie;
c
Escuela de ingeniería civil en
informática, Universidad de Valparaíso, Chile
ABSTRACT
The obesity pandemic is represented by a discrete-time Hopfield
Boolean network embedded in continuous-time population
dynamics. The influence of the social environment passes through
a system of differential equations, whereby obesity spreads by
imitation of the most influential neighbors, those who have the
highest centrality indices in the network. This property is called
“homophily.”Susceptibility and frailty are redefined using network
properties. Projections of the spread of obesity are validated on
data collected in a French high school.
KEYWORDS
social networks; obesity
pandemic spread modeling;
frailty modeling; public
health policy against obesity
1. Introduction
Frailty designates the set of co-morbidities associated with diabetes of type 2 such
as obesity, retinitis, nephritis, cardio-vasculardisease,brainstroke,andvascular
dementia. Obesity is due to an excessive accumulation of fat in the adipose tissue.
The World Health Organization (WHO) estimates that the prevalence of obesity
has tripled between 1980 and 2000 (IASO, 2000). Myers and Rosen (1999)and
Darmon (2009) argue that the genetic ground is worsened by the stigmatization or
on the contrary by the imitation of alimentary habits. Tunstall-Pedoe (2003)
showed that the prevalence of obesity in Europe has increased from 10% in
1992 to 20% in 2012 for men, and from 10% to 25% for women. Christakis and
Fowler (2007) showed that obesity spreads through social life. Cohen-Cole and
Fletcher (2008) suggested that an exogenous factor could favor the diffusion of
obesity (Sathik and Rasheed, 2011;Demongeotetal.,2016a). Identifying this
factor involves education, work, and residence. It is favored by the participation
in alcohol and smoking cessation programs and by the regular comsumption of
fast food. The difficulty in quantifying the dependence on the environment has led
medical and social workers to use simple scores of risk, such as the Quality of Life,
Obesity, and Dietetics rating scale (Ziegler et al., 2005). This score is consistent
with the general International Classification of Functioning, Disability, and Health
CONTACT Jacques Demongeot Jacques.Demongeot@yahoo.fr Laboratory AGEIS, University Joseph Fourier,
Faculty of Medicine, Domaine de la Merci, 38700 La Tronche, France.
MATHEMATICAL POPULATION STUDIES
2017, VOL. 24, NO. 4, 219–245
https://doi.org/10.1080/08898480.2017.1348718
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(WHO, 2013,2016). Here, we model the role of some determinants in the spread
of obesity. First, we revisit factors arguably involved in obesity (section 2). Then,
we build homophilic graphs relating individuals for their risk with respect to
obesity (section 3). We present simulated data articulating demography and the
spread of obesity in section 4,andacontagionmechanisminsection 5, showing
the necessity to improve the simulated model to perform realistic predictions
about the public health policies liable to reduce the spread of obesity.
2. Determinants of obesity
Table 1 synthesizes the review done by Jelassi et al. (2014) and Demongeot
et al. (2016a) of determinants of obesity.
3. Method
3.1. Homophilic Hebbian graphs
The social network with interaction graph Gis characterized by the total number
k
i
of neighbors connected to individual i. We consider random links, scale-free
links (the total number of links to a node follows a power law); small-world
networks (most nodes have few neighbors but can be reached from every node
of the network through a small number of hub nodes (nodes having many links);
and three empirical networks, two (1 and 2) taken from Christakis and Fowler
(2007) and one (3) from Demongeot et al. (2016a). Figure 1 presents simulated
examples of the first three networks, and the three empirical networks.
We call “homophily”the tendency of an individual to have links to indivi-
duals with similar attributes and none with individuals having no attributes in
common. We combine homophily with what we call a Hebbian rule, consisting
in deleting or strengthening activating or inhibitory links between individuals.
3.2. The homophily algorithm
The interaction graph Gcontains nodes and oriented links. It is equipped
with a distance dmeasuring the similarity between nodes iand j.
(1) First step: the Hebbian rule. An individual at node i(i=1,...,N)attimetand
at state x(t,i)iseitherS(for susceptible), W(for overweight), or O(for
obese). His age is A(t,i); his body adiposity index B(t,i); the total number of
his relatives F(t,i). He has C(t,i) friends; he can access to P(t,i) green areas
and M(t,i) supermarkets; he has spent L(t,i) hours at home and S(t,i)hours
practicing sport during the last 24 hours. The attributes (x(t,i), B(t,i), F(t,i),
C(t,i), P(t,i), M(t,i), L(t,i), S(t,i)) are gathered in the state vector V(t,i). We
compute a correlation with the vector V(t,j) of individual j, by calculating at
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Table 1. Models of obesity.
Models of Obesity
Characteristics Comparison
Citation Evangelista et al. (2004) Christakis and Fowler (2007) González-Parra et al.
(2010a)
Type ordinary differential
equations
static network stochastic ordinary
differential equations
Purpose study the role of peer
pressure on becoming
fast food eater and its
effect on the weight of an
individual
study whether weight gain of
an individual is associated with
weight gain of friends, siblings,
neighbours, etc.
investigate the dynamics
of an obese population
within a varying social
environment
State variables sex, age, ethnicity,
education, income
sex, age, education, body mass
index of individuals of different
generations and their friends,
parents, and neighbors
unhealthy eating habits
Scale population individuals population
Citation González-Parra et al.
(2010b)
Taramasco, (2011) Bourisly (2013)
Type partial differential
equations
individual-based model agent-based model
Purpose model the correlation of
the development of
obesity with age and time
for forecasting
study the spread of obesity in
either a given static or a
dynamic social structure,
considering the double inter-
individual and individual-
environment
interactions
model the obesity
epidemic and create a
decision support system
to reduce obesity
State variables age obesity state, tolerance to
external influence
obesity state
Scale population population and individuals individuals
Citation Bahr et al. (2009) Burke and Heiland (2006) Hammond and Epstein
(2007)
Type network-based model agent-based model agent-based model
Purpose simulate the spread of
obesity through social
networks and advise
management
explore factors of obesity
prevalence by considering food
price and metabolic
heterogeneity
find simple, individual-
based, nonprice
mechanisms of the
obesity epidemic
State variables obesity state obesity state obesity state, sex,
physical activity, diet
Scale individuals individuals individuals
Citation Shoham et al. (2012) Schumm et al. (2013) Shoham (2015)
Type stochastic actor-based
model
individual-based model: a
contact network and a
susceptible-latent-infected-
recovered model
stochastic actor-based
model with personal
networks
Purpose identify social contagion
by network dynamics
(friendship selection
based on homophily and
structural characteristics)
and social influence
study the effect of vaccination
campaigns with limited
resources, through a stochastic
susceptible-latent-infected-
recovered Poisson process
identify social contagion
by network dynamics
(friendship selection
based on homophily and
structural characteristics)
and social influence
State variables adolescent body mass
index, sex, education,
race, income, screen time,
practice of sports
disease state variables adolescent body mass
index, sex, education,
race, income, screen
time, practice of sports
Scale individuals individuals individuals
(Continued)
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Table 1. (Continued).
Models of Obesity
Characteristics Comparison
Model Fonseca et al. (2009) Gregori et al. (2012) Mulvaney-Day et al.
(2012)
Type statistical study statistical study of a social
network
statistical study
Purpose examine the correlation
between the emotional,
behavioural, social
aspects, and missing
values for body mass
index
explore factors explaining the
increase of obesity prevalence
by considering food price and
metabolic heterogeneity
identify factors of
individual health with
respect to healthy eating
choices
State variables weight and height, body
image, dieting for weight
loss, physical activity,
perception of academic
achievement, irritability,
nervousness and
depressive mood,
absence of parents
body mass index, family
composition, father and mother,
mother’s perception of the
child’s weight, physical activity
of child, father and mother, time
spent watching television or
playing video games, use of
social networks; leisure time,
dietary habits of peers, income
per year
eating behaviors (time,
cost, restaurant policies,
social networks),
psychological factors
associated with hunger,
food knowledge versus
food preparation know-
how, reaction to physical
experiences, perceptions
of food options, delayed
gratification, and
subjectivity
Scale population population population
Citation Ejima et al. (2013) Mir et al. (2012) de la Haye et al. (2011)
Model Type ordinary differential
equation model
linear regression stochastic actor-oriented
model
Purpose model the obesity
epidemic accounting for
social contagion and non-
contagious hazards of
obesity, to compare the
effectiveness of
interventions
investigate long-term effects,
during adolescence
or early adulthood, of peers on
individual weight outcomes
determine whether
weight-based similarities
among adolescent
friends result from a
social influence or not
State variables age body mass index, age, sex, race,
and education
peer measure variables: number
of friends and peers in the same
and class, percentage of
overweighted friends,
percentage of friends at risk of
being overweighted and their
mean body mass index, others:
smoking status, hours of
television watched per week,
exercise and sport habits, total
number of siblings, birth
weight, birth order, breastfed or
not, composition of the family,
parents’occupations and
obesity status, family income
height, weight, sex,
ethnicity, pocket money,
names of best friends
Scale population individuals individuals
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each time kT (k=1,2,…) the average cross-correlation between the V
componentsduringthetimewindow[(k-1)T,kT[. The Hebbian rule con-
sists of deleting any link between uncorrelated nodes and establishing a link
between correlated nodes. This process is slower than the homophily
dynamic, because Tis chosen higher than the average updating time of
the homophily rule and most of the components of Vvary slowly.
Figure 1. Possible initial networks: random (a), scale-free (b), small world (c), and empirical 1 (d),
empirical 2 (e), and empirical 3 (f).
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(2) Second step: the homophily rule for links. At time t, L
a,b
(t)isthetotal
number of heterophilic links from a node in state ato a node in state b, L
a
(t) the total number of links to nodes of type a,andL(t) the total number
of links. The relaxation time is a random variable τ<T, which corre-
sponds to the lapse of time during which links are established or deleted.
At t=t
0
, generate the random value τfrom an exponential distribution of
parameter 1/β, where β>0 represents the expected relaxation time; at t=t
0
+τ,
choose a total number M(0<M≤N) of nodes in G. For each node iof these M
nodes, define the initial state x(0,i), its total number k
i
of links exiting from i,
its tolerance h
i
to the difference between the state of iand those of his
neighbors, by drawing from a probability distribution gon [0,1]. For k
i
=
0, connect ito jby:
●choosing a node jat random among N–1 other nodes;
●creating a link from ito jwith probability h
id(t;i,j)
, where d(t;i,j) is the
distance between iand jat time t, among the three possible values 0, 1,
or 2:
dt;i;jðÞ¼0;if xt;iðÞ¼xt;jðÞ;
¼1;if xt;iðÞÞxt;jðÞand xt;iðÞ;xt;jðÞ
fg
¼S;W
fg
or W;O
fg
;
¼2;if xt;iðÞÞxt;jðÞand xt;iðÞ;xt;jðÞ
fg
¼S;O
fg
:
(1)
For k
i
≥1, connect or disconnect ito jby:
●choosing a node jamong the set V
i
of neighbors of iwith the probability
1/k
i
.V
i
is composed of iand of its neighbors kdifferent from i:
k2Vii¼Vini
fg
;
●r(t;i,j) is the similarity distance between nodes iand jat time t.The
link between iand jis created with the probability h
ir(t;i,j)
(with a
proportion γof uni-directional and (1-γ) of bidirectional links), and
is deleted with the probability (1-h
ir(t;i,j)
), where the similarity dis-
tance ris defined by:
rt;i;j
ðÞ
¼dt;i;j
ðÞ
;if ct;i;j
ðÞ
¼0;
rt;i;jðÞ¼αdt;i;jðÞþð1αÞct;i;jðÞ;if ct;i;jðÞÞ0;
(2)
where the indirect distance c(t;i,j) between iand jat time tis defined by:
ct;i;jðÞ¼
Xk2Vi
i
dt;i;kðÞ
kj1:(3)
If the link between iand jis deleted, a node kis chosen in G\(V
i
∪V
j
) together
with a link from ito kwith probability:
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Pt;i!kðÞ¼
fdt;i;kðÞðÞnxt;i;kðÞ
hdt;i;kðÞ
i
Zt;iðÞ ;(4)
where Z(t;i)=Σ
l∈G\(Vi∪Vj)
f(d(t;i,l)) n
x(t;i,l)
h
id(t;i,l)
and n
x(t;i,l)
is the total
number of nodes in V
l
occupying the same state as i, which means either
n
x(t;i,l)
=n
S,l
, if the state of lis susceptible, with n
S,l
the total number of
susceptible in V
l
,orn
x(t;i,l)
=n
W,l
, if the state of lis overweight, with n
W,l
the
total number of overweight in V
l
,orn
x(t;i,l)
=n
O,l
, if the state of lis obese,
with n
O,l
the total number of obese in V
l
. The linkage function ftakes three
possible values corresponding to three different cases:
1:fðdt;i;kðÞÞ¼1;if dt;i;kðÞ¼0;¼0 elsewhere;
2:fðdt;i;kðÞÞ¼1;if dt;i;kðÞ¼0or1;¼0 elsewhere;
3:fðdt;i;kðÞÞ¼1;if dt;i;kðÞ¼0;1;or 2:
(5)
In the homophily dynamic, these three cases represent an endogenous influ-
ence (case 1), an exogenous homogeneous influence (case 2), or an exogenous
heterogeneous influence (case 3).
(3) Thirdstep: homophilyrule for nodes. For each node kin V
i
, the state x(t,k)
changes with the transition rule:
"a2S;W;O
fg
;Pxtþ1;kðÞ¼aðÞ¼
na;t;ihdt;i;kðÞ
i
Yt;iðÞ ;(6)
where Y(t;i)=Σ
b∈{S,W,O}
n
b,t,i
h
id(t;i,k)
and n
b,t,i
is the total number of neighbors
of iin state bat time t.
(4) Fourth step: generate a relaxation time τand go to step 2.
(5) Fifth step: stop when the contagion graph no longer changes after an
iteration.
In the homophily scheme, individuals interact with those alike. The degree of
homophily depends on the tolerance coefficient hand on the linkage function f.
Other models presented in Table 1 includeananalogueofthetolerance
coefficient.
4. Results
4.1. Equilibrium of the mean homophily coefficient
Under the homophily rule, the network converges to an equilibrium for both
nodes and links for any initial network. In Figures 2 and 3,thestates“obese”
(14.5% of the population), “overweight”(31.9%), and “normal weight”(53.6%)
MATHEMATICAL POPULATION STUDIES 225
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follow the distribution of the body mass index among the French population in
2009 (Obepi-Roche, 2009). The tolerance is 0.25, which means that an individual
expects one fourth of his neighbors in a state different from his, and the linkage
probability, called the “endogenous influence,”corresponds to the case 1 of Eq.
(5). Figure 2 shows examples of directed empirical networks. Each has 1,000
nodes, with 60% odds to have uni-directional links. At each iteration, M=500
nodes have their directions changed. The weight αof the similarity distance
equals 0.85. The two-dimensional attraction-repulsion Fruchterman-Reingold
algorithm (1991) determines the nodes in such a way that all links are of equal
length and cross one another the least as possible. In Figure 2,theinitial
empirical network 1 has a mean in-degree; that is, a mean total number of
links to a node, equal to 1.220 and the empirical network 2 has a mean in-degree
equal to 3.759. Network 2 displays more asymptotic segregation of individuals
by types than Network 1, but its mean homophily coefficient H(t), defined by:
HtðÞ:PaS;W;O
fg
La;atðÞ
LtðÞ ;(7)
converges more slowly to the equilibrium. This behavior holds true for the
mean tolerance coefficient equal to 0.1 on the left-hand side of Figure 3 and
equal to 1 on its right-hand side.
Figure 2. Simulated graphs representing homophily dynamics. The two rows show a sequence of
graphs starting from an initial empirical network with a mean total number of ingoing edges
1.22 for the first row and 3.759 for the second row. The transient network after 40 iterations is
represented in (b) for the first row and in (e) for the second row. The asymptotic states (observed
after 400 iterations) are represented in (c) for the first row and in (f) for the second row.
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Figure 4 shows empirical networks associated with obesity for the French
high school of Jœuf (in the French département of Meurthe-et-Moselle), where
pupils were between 10 and 14 years of age (Demongeot and Taramasco, 2014;
Demongeot et al., 2015a, b). Figure 4 top shows that the total number of obese
increases, while they were gradually segregated as in Figure 3.Figure 4 middle
presents a group of obese or overweight pupils (red points of small size) having
only few friends (two on average) and a group of obese or overweight pupils
(red points of large size) having more friends (eight on average). The second
group is the target of preventive education, such as personalized diet
(Demongeot et al., 2016a). The simulation on Figure 4 bottom shows that
tuning the tolerance hallows the fit for different behaviors such as those
recorded in the sixth grade (pupils of age 10 to 11) and the ninth grade (pupils
of age 13 to 14) in Jœuf high school.
4.2. Age as contributing to frailty among obese people
DOPAMID is a demographic projection software (Teymoori et al., 2010;
Demongeotetal.,2013a,b).InFigure 5, each node is either a youth or an
adult. The arrows indicate possible changes of category. We consider the group
G
1
of normal-weight people, the group G
2
of persons at risk of complications,
and the group G
3
of frail elderly of over 75 years of age.
The model DOPAMID relies on the age- and sex-specific distribution of the
population, on life tables, on the fertility age distribution, on the composition of
families, on the states of individuals with respect to weight, and on the frailty of
Figure 3. Left-hand side: the mean homophily coefficient Hconverges to the equilibrium. The
mean tolerance is 0.1, and the initial network is indicated below the curves. Right-hand side:
same trajectory as on the left-hand side, with the mean tolerance equal to 1.
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individuals presenting co-morbidity due to obesity or ageing. Starting from
national statistics in 2012, the population is projected until 2050. The sex ratio
at birth is generated from a Bernoulli law of 0.487 girl for 0.513 boy, and the
initial Iranian population size is 76,420,000 (World Bank, 2012). Figure 6 top
compares our projections of the Iranian population in the years 2020 and 2050
to the projections by U.S. Census simulator (Census, 2016). The close fit
validates the software DOPAMID.
Figure 4. Top: friendship network relating obese, normal weight and overweight individuals to
their friends in two classes of the French high school of Joeuf (in the French department of
Meurthe-et-Moselle): a sixth-grade class of 10–11 years of age on the left-hand side and a ninth-
grade class of 13–14 years of age on the right-hand side. Middle: histogram of the total number
of friends for the pupils of the network, gathering the four classes of the French high school of
Joeuf (10–14 years of age). Bottom: two simulations using Eq. (6), with a tolerance h=exp(1/θ), θ
being a stochasticity parameter equal to 1 on the left-hand side and to 2 on the right-hand side.
The higher θ, the closer the probability of Eq. (6) to the majority rule.
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Harati et al. (2009) shows that the proportion in the Iranian population of
diabetic people of type 2 is 3.5% in normal weight, 6.4% in overweight, and
14.3% obese. From initial conditions in Hosseinpanah et al. (2009), the projec-
tion yields the proportion of obese people and the proportion at risk of type 2
diabetes at each age. Demographics are combined with the social network in the
first step of the algorithm. Figure 6 bottom shows helping rates H
i
,whicharethe
ratios of the total number of potential helpers between 50 and 59 years of age,
over the total number of individuals frail but moderately dependent, constitut-
ing the cluster C
1
of the group G
3
of frail elderly of over 75 years of age. The
cluster C
2
corresponds to frail and completely dependent people and the cluster
C
3
to individuals who are frail, completely dependent, and needing help at least
twelve hours a day. Figure 6 shows a rapid decrease, which corresponds to an
unstatisfied demand for care, of H
2
and H
3
at the late 2040s.
5. Contagion in social diseases
5.1. Influence of the duration of contact
We introduce an integral term to reflect the intensity of contagion in a social
network depending on the duration of contact T(Demongeot et al., 2013a,b;
Magal and Ruan, 2014). Bernoulli (1760), d’Alembert (1761), Hamer (1906),
Ross (1910,1915,1916), and Brownlee (1915) formalized the infectious rate to
be proportional to the product SI of the total numbers of susceptible Sand
infected I. After Delbrück (1940) and Bailey (1950), we consider Sand Ito be
random variables. We now ignore the demographics and consider that people
are either of normal weight or obese. At time t-τ, for a duration τin [0,T], one
person among the S(t-τ)=k+1 susceptible may become obese and then increase
Figure 5. Exchanges between 7 subgroups of the young and adult subpopulations among N
individuals being either in state S(normal weight), W(overweight), O(obese), or F(frail). The size
of the young sub-population is N1 and the size of the adult sub-population N2.
MATHEMATICAL POPULATION STUDIES 229
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the total number of obese O(t-τ)=N-k-1during[t, t+dt] by 1, with conditional
probability:
dt T
0βkþ1ðÞNk1ðÞdτ;(8)
where at time t, S(t) is the total number of normal-weight people, O(t) the
total number of obese people, βis the contagion intensity, and Nthe constant
total population size.
The unconditional probability is:
dt T
0βkþ1ðÞNk1ðÞPðSðtτÞ¼kþ1;OðtτÞ¼Nk1Þdτ:
(9)
The alternative is that this susceptible person does not become obese, and
this occurs with the unconditional probability
ð1dt T
0βkNkðÞdτÞPðSðtτÞ¼k;OðtτÞ¼NkÞÞ:(10)
By summing the probabilities of Eq. (9) and (10), and neglecting the transitions
from obese to normal or overweight:
PStþdtðÞ¼k;OtþdtðÞ¼NkðÞ¼dt T
0ðβkþ1ðÞ
ðNk1ÞPðSðtτÞ¼k;OðtτÞ¼NkÞÞdτ
þð1dt 
T
0βkNkðÞdτÞPðSðtτÞ¼k;OðtτÞ¼NkÞ;(11)
Figure 6. Left-hand side top: DOPAMID simulation of the Iranian age pyramids for 2020 and
2050. Left-hand side bottom: U.S. Census simulation of the Iranian age pyramids for 2020 and
2050. Right-hand side: helping rate Hi for the cluster Ci of the group G
3
of frail elderly of over 75
years of age; for example, H1 equals the total number of potential helpers between 50 and 59
years divided by the size of the cluster C1.
230 J.DEMONGEOTETAL.
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and then the transition rate from normal weight to obese is:
PStþdtðÞ¼k;OtþdtðÞ¼NkðÞPStðÞ¼k;OtðÞ¼NkðÞ
¼βdt 
T
0ðkNkðÞPðSðtτÞ¼k;OðtτÞ¼NkÞþ kþ1ðÞ
ðNk1ÞPðSðtτÞ¼kþ1;OðtτÞ¼Nk1ÞÞdτ:(12)
By denoting Efor expectation, assuming independence between Oand S
(which is only approximately true) we have:
EStðÞðÞðÞ
0¼βT
0EðSðtτÞÞ EðOðtτÞÞdt;(13)
and by equating Xto E(X) for all variables X:
S0tðÞ¼βT
0SðtτÞOðtτÞdτ:(14)
Eq. (8) and (14) constitute an integro-differential system, with explicit duration
of contact T. It was used in the modeling of epidemic contagion and in chemical
reactions (Delbrück, 1940; Bartholomay, 1958a,1958b,1959;Demongeot,1977;
Rhodes and Demetrius, 2010; Britton, 2010) and in agent-based models of
epidemics (Gillespie, 1970; Demongeot et al., 2015a).
5.2. Influence of contact saturation
In Figure 7, the contagion graph is incompatible with the interaction graph. That
is why we extend the quadratic contagion term to a rational fraction, allowing for
the introduction of a continuous-time change of state for the nodes and a
nonlinear hybrid-time rule (continuous-time for the decision, discrete-time
for the spread). To do this, we solve the inverse problem of finding the closest
interaction graph tothe observed contagion graph such as the one in Figure 7.At
time t, S(t) is the total number of susceptible individuals and I(t) the total
number of infected. The quadratic contagion term βS(t)I(t)isjustifiedonlyif
encounters are independent of each other, and if the total numbers of susceptible
and infected individuals are both small. Otherwise, if the total number of
susceptible I(t) is small compared to that of infected S(t), the contagion term
saturates because encounters between many susceptible and few infected are
limited by the total number of infected who cannot infect more than βI(t)
susceptible duringa unit of time: then the simplest way to saturate the contagion
term in this case is to use the rational fraction bStðÞ
KþStðÞItðÞ,withβ=b/K.Likewise,
if S(t)issmallandI(t) large, the saturating term is bS tðÞ ItðÞ
LþItðÞ. Both cases are
captured by the contagion term bStðÞ
KþStðÞ
ItðÞ
LþItðÞ, which reduces to βS(t)I(t), with
β=b/KL,ifS(t)andI(t) are both small. If S(t)andI(t) are both large, the
contagion term equals β, which corresponds to the maximal contagion rate in
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the population. Then contagion is governed by:
x0tðÞ¼ bx tðÞytðÞ
Kþxt
ðÞðÞ
Lþyt
ðÞðÞ
;y0tðÞ¼ bx tðÞytðÞ
Kþxt
ðÞðÞ
Lþyt
ðÞðÞ
μytðÞ;(15)
where x(t) represents the total number of susceptible and y(t) the total
number of obese in the social network at time t.Forthesakeofsimpler
calculation, we take b=K=L=1, X¼x
1þx;and Y¼y
1þy;which leads us to
Figure 7. Top: the empirical network 3 of Figure 4 at the start of the contagion. The red arrows
represent the contagion process. Bottom: a contagion graph adapted from the Centers for
Disease Control and Prevention (2003), where individuals are represented by successive small
circles corresponding to the dates at which they are “infected.”
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X0ðtÞ¼XðtÞYðtÞð1XðtÞÞ2;
Y0ðtÞ¼XðtÞYðtÞð1XðtÞÞ2μYðtÞð1YðtÞÞ:
(16)
Eq. (16) resembles a potential-Hamiltonian equation, which is (Demongeot
and Waku, 2012a,b):
X0ðtÞ¼@PðtÞ
@XðtÞþ@HðtÞ
@YðtÞ;
Y0ðtÞ¼@PðtÞ
@YðtÞ@HðtÞ
@XðtÞ;
((17)
where Pand Hare polynomials in Xand Y:
PX;YðÞ¼μY2
2X3Y3
6μY4
4þX5Y5
20 X2Y2XYðÞ
2
HX;YðÞ¼XY XþY
2þX2Y2 2XY
2þXY X3þY3
4:(18)
Because the rational fractions X¼x
1þxand Y¼y
1þyare always less than 1, by
neglecting the terms of order 3 and over, we have:
PX;YðÞμY2
2and HX;YðÞ0;(19)
and Eq. (17) works like a quasi-dissipative potential system with principal
potential part (Krasnoselsky, 1968). We denote (x
ss
,y
ss
) the stationary states
of Eq. (15). The Jacobian matrix is:
Jss ¼
yss
ð1þxssÞ2ð1þyss Þ
xss
ð1þyssÞ2ð1þxss Þ
yss
ð1þxssÞ2ð1þyss Þ
xss
ð1þyssÞ2ð1þxss Þμ
!
(20)
as y
ss
=0,
Jss ¼0xss
1þxss
0xss
1þxss μ
 (21)
with the characteristic equation:
λðλxss
1þxss
ðÞ
þμÞ¼0:(22)
The stationary state (x
ss
,y
ss
) is a stable node in xin the sense of Lyapunov
and asymptotically stable in yif µ>1.
5.3. Transition rule based on a continuous-time decision process
We consider now that links are permanent, and that the probability for the
node kto be in state x(t+1,k)=1 (obese) at time t+1, knowing that the nodes of
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its neighborhood V
k
(the set of nodes linked to node k) are in states x(t,i), i∈V
k
,
follows a rule similar to that defined in Eq. (6):
Pðxtþ1;kðÞ¼1jxt;iðÞ;i2Vk
ðÞðÞ¼
exp Ht;k
ðÞðÞ
1þexp Ht;kðÞðÞ
;(23)
where
Ht;kðÞ¼ln X
i2Vk
xt;iðÞhdði;kÞ
!
ln X
i2Vk
1xt;iðÞðÞhdði;kÞ
!
;(24)
and V
k
is the set of nodes ilinked to kin the interaction graph, with d(i,k)=1/θ.θ
is a stochasticity parameter. The rule defined in Eq. (23) is deterministic if θ=0
(the probability is null to be obese); it is analogue to the rule defined in Eq. (6) if
θ=1; it is a majority rule if θtends to infinity. Eq. (23) is a stochastic equation
similar to Eq. (6): the higher θ, the closer to a majority rule is the probability of
being obese. With n(t,k) denoting the cardinal of i2Vk,n(1,k,t) the total
number of obese neighbors of k,andn(0,k,t) the total number of neighbors in
normal weight or overweight, then the rule defined in Eq. (6) can be written, if
θ=1:
Pðxtþ1;kðÞ¼1ðÞjxt;iðÞ;i2Vk
ðÞÞ¼
n1;t;kðÞ
n1;t;kðÞþn0;t;kðÞh¼1n0;t;k
ðÞ
n
1n0;t;kðÞ
nþhn1;t;kðÞh
n
¼
1þP
j2VK;xðt;jÞ¼0
vkj
1þhP
j2VK;xðt;jÞ¼1
hvkj þ1þP
j2VK;xðt;jÞ¼0
vkj
;(25)
where wki ¼1
nand vkj ¼1
n:
We define O
k
:= Σ
i∈Vk;x(t,i)=1
w
ki
and N
k
:= Σ
j∈Vk;x(t,j)=0
w
kj
.If0<O
k
< 1, then:
1OkexpðOkÞ Ok2
2;if 1<Nk<0;then 1 NkexpðNkÞNk2
2:
By neglecting second-order terms:
Pðxtþ1;kðÞ¼1ðÞjxt;iðÞ;i2Vk
ðÞÞ
¼
exp Pj2Vk;xt;jðÞ¼0vkj

exp Pj2Vk;xt;j
ðÞ¼
0vkj

þexp Pi2Vk;xt;i
ðÞ¼
1wki þlnh

:(26)
This is the same formula as in Hopfield networks. For example, if V
k
contains
only two nodes, one obese iand the other jof normal weight:
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Pxtþ1;kðÞ¼1ðÞjxt;iðÞ;i2Vk
ðÞðÞ¼p1tþ1;kðÞ
¼exp wki þvkj þq

1þexp wki þvkj þq

¼exp qðÞ
1þexp qðÞ
;(27)
where q=-lnh. p
1
(t,k), estimated by Pi2Vkwkixt;iðÞ¼
1
nPi2Vkxt;iðÞ;repre-
sents the probability of a member of V
k
to be obese. Its estimator is the
proportion of obese members of V
k
. By differentiation:
@p1t;kðÞ
@tp1tþ1;kðÞp1t;kðÞ¼
exp Pi2Vkakixt;iðÞþq

1þexp Pi2Vkakixt;iðÞþq

p1t;kðÞ;
(28)
where
Xi2Vkakixt;iðÞ¼
Xi2Vk wkixt;iðÞþvki 1xt;iðÞðÞðÞ
¼Xi2Vkðxt;iðÞþxt;iðÞ1ðÞÞ
¼1
nXi2Vk 2xt;iðÞ1ðÞ:(29)
If ∑
i∈Vk
a
ki
x(t,i)+qis small,
@p1t;kðÞ
@t1
41þXi2Vkaki xt;iðÞþq

2Xi2Vkaki xt;iðÞq

p1t;kðÞ:(30)
If q=2, which corresponds to h=e
−2
≈0.13:
@p1t;kðÞ
@t1
4n23nþXi2Vkð2xt;iðÞ1

Xi2Vkð12xt;iðÞ

p1t;kðÞ:(31)
Hence, with p
1
denoting p
1
(t,k):
@p1t;kðÞ
@t3þp11p1
ðÞðÞp1þ1p1
ðÞðÞ
4p1
¼ðð14
3Þp1þ1
4bð1p1ÞÞð3ð14
3Þp1bð1p1ÞÞ  2
3p1ðð14
3Þp1
þbð1p1ÞÞ:(32)
By neglecting the terms of second order in p
1
:
@p1t;kðÞ
@t1
4ð2p1t;kðÞþ21p1t;kðÞðÞ
¼Lap
1þbð1p1
ðÞÞKap1þbð1p1
ðÞðÞÞ;(33)
which corresponds to the continuous-time decision process (Demongeot and
Volpert, 2015; Banerjee et al., 2016) where K=3, L=1/4, a=-1, and b=1. The
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unique stable stationary state corresponds to p
1
=0.5. If overweight people are
in proportion 1-p
1
-p
2
, where p
2
(t,k) is the proportion of individuals in
normal weight, then the distribution between obese, overweight, and normal
weight is governed by:
@p1ðt;kÞ
@t¼Lðap1þbp2ÞðKðap1þbp2ÞÞ;
@p2ðt;kÞ
@t¼Lðcp1þdp2ÞðKðcp1þdp2ÞÞ:
((34)
Eq. (34) has six stationary points represented on Figures 8 and 9:E
0
=(0, 0),
E
1
=(K/a, 0), E
2
=(0, K/d), E
3
=(bK/(bc-ad), aK/(ad-bc)), E
4
=(0, bK/(bc-ad)),
and E
5
=(p
1(5)
,p
2(5)
), where p
1(5)
and p
2(5)
are solutions of:
ap1þbp2¼Kand cp1þdp2¼K:(35)
The Jacobian matrix J
3
associated with the fixed point E
3,
is:
J3¼k1K2ap12bp2
ðÞk2ðK2ap12bp2Þ
k3K2cp12dp2
ðÞk4K2cp12dp2
ðÞ

;(36)
where k
1
=La, k
2
=Lb, k
3
=Lc, and k
4
=Ld.
Then the characteristic equation of Jis:
λ2k1K2ap12bp2
ðÞþk4K2cp12dp2
ðÞðÞλ
þk1k4k2k3
ðÞK2ap12bp2
ðÞK2cp12dp2
ðÞ
¼0 (37)
and the eigenvalues λof Jverify:
Figure 8. Trajectories of Eq. (34) in the plane (p1, p2) when c>a and d>b.
236 J.DEMONGEOTETAL.
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2λ¼k1K2ap12bp2
ðÞþk4K2cp12dp2
ðÞðÞ
ð k1K2ap12bp2
ðÞþk4K2cp12dp2
ðÞðÞ
2
4k1k4k2k3
ðÞK2ap12bp2
ðÞK2cp12dp2
ðÞðÞ
1
2:ð38Þ
The stationary point E
0
is always unstable. E
1
is a stable node if and only if
c>a, and E
2
is a stable node if and only if d>b(Figure 9). If a>c, E
3
is
surrounded by a limit cycle, and, if b>c, E
4
is surrounded by a limit cycle
(Figure 9 left-hand side).
If K-ap
1
-bp
2
=K-cp
1
-dp
2
=0, which implies (a-c)(d-b)>0, for E
5
to be
positive, the Jacobian matrix Jbecomes:
J¼k1Kk2K
k3Kk4K

:(39)
The eigenvalues of Jare:
2λ¼Kk1þk4
ðÞk1þk4
ðÞ
24k1k4k2k3
ðÞ

1
2

¼KL aþdðÞaþdðÞ
24ad bcðÞ

1
2

:(40)
If a>cand d>b, E
5
isastablenode(Figure 9 left-hand side), E
5
is a
saddle, and E
1
and E
2
are stable foci (Figure 9 right-hand side) if and only if
a≤cor d≤b.
Figure 9. Left-hand side: trajectories of Eq. (34), obtained by Zweigmedia’s(2016) method,
showing the stability of E5, which is a node. Right-hand side: trajectories of Eq. (34), showing
the stability of the stationary points E1 and E2, which are nodes, and the instability of the
stationary points E0, which is unstable, and E5, which is a saddle.
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5.4. Nonlinear interactions and phase transition
We solve here the open problem of finding a hybrid rule using the threshold state
transition in Hopfield’s(1982) model. This model involves only pairwise contacts,
ignoring possible additional effects due to the presence and mutual interaction of
more than two individuals during the contagion. We introduce these effects by
defining nonlinear n-uple interactions (Demongeot and Sené, 2011). We run the
model alternatively as spatial Markovian with a local spatial influence of a group
on one individual or with renewal distributed over space (Demongeot and Fricot,
1986), with distant spatial influence between individuals connected through the
Internet (Barthélémy et al., 2010). The main problem of spatial influence comes
from the possibility of having various stationary distributions of the states in the
social network, corresponding to various asymptotic behaviors (in time or in
space). This phenomenon depends on some critical nodes, which are nodes i
having only pending nodes in their neighborhood V
i
. We show that this phenom-
enon cannot occur in the case of weak interactions whether they are activating or
inhibitory.
Assume that the nodes are localized on a finite regular subset C
n
:C
n
=[-(k-1),
k]
2
of ℤ2,wheren=2kis an even integer. These nodes have only two possible
states, 1 for obese or overweight and 0 for susceptible. C
n
has n
2
elements (or
nodes), with 4(n-1) elements located on its boundary defined by:
FCn¼x2Cn=Vx\CnCÞ;

;(41)
where V
x
is the neighborhood of xin ℤ2,definedbyV
x
={z∈ℤ2|d(z,x)≤1}, with
dthe L
1
distance of ℤ2,andC
nC
the complementary set of C
n
in ℤ2.∂C
n
is the
boundary of C
n
, equal to the boundary FC
nC
of C
nC
,andC
n
°theinteriorofC
n
:
C
n
°=C
n
\FC
n
.Thecardinalof∂C
n
is 4n,thecardinalofFC
n
is 4(n-1), and the
cardinal of C
n
°isn
2
-4(n-1).
Proposition: If A is the maximal subset of C
n
where the nodes are in state 1,
assume that the probability to observe A knowing that the nodes of ∂C
n
in state
1 belong to B
n
⊂∂C
n
is a Boltzmann measure equal to:
Pnððxi
ðÞ¼1;i2A;xi
ðÞ¼0;i2CnnAÞ=ðxj
ðÞ¼1;j2Bn;xj
ðÞ¼0;j2@CnnBn
ð ÞÞÞ ¼
expðP
i2A
wiixðiÞþ P
i;j2A
wijxðiÞxðjÞþ P
i2A;k2Bn
wiixðiÞxðkÞÞ
Zn
;(42)
where
Zn¼Xi2DCn expðXi2DwiixiðÞþXi;j2DwiixiðÞxjðÞÞþ
Xi2D;k2Bn
wikxiðÞxkðÞÞ:
(43)
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The quantity defined by:
UBn
nDðÞ¼
Xi2DwiixiðÞþXi;j2Dwij xiðÞxjðÞÞþ
Xi2D;k2BnwikxiðÞxkðÞ (44)
is called the potential of D. If ∀i,j ∈A, k ∈B
n
,w
ii
=u
0
,w
ij
=w
ik
=u
1
, with inf(u
0,
u
0
+4u
1
)=βand sup(u
0,
u
0
+4u
1
)=αsufficiently small, then P
n
converges to a
unique equilibrium measure µ, when n tends to infinity, independently of the
choice of the sequence {B
n
}
n∈ℕ
.
Proof: C
n
consists either in sets A, whose interiors A° or the interiors of their
complementary sets A°
C
in C
n
have O(n
2
)nodes,orinsetsB,whoseinteriorsB°
and the interiors of their complementary sets B°
C
in C
n
have at most O(n)nodes.□
In the first case of sets Aand when all nodes of the boundaries ∂C
n
of C
n
are in
state 1, a node iin state 1 belonging to the interior A°ofAcontributes of w
ii
+Σ
j∈Vi
w
ij
=u
0
+4u
1
to the potential U
∂
n
(A)ofA;andanodekin state 1 belonging
to the boundary ∂Aof Acontributes of w
kk
+Σ
j∈Vj
w
kj
+Σ
j∈∂C
n
w
kj
≤u
0
+4u
1
to the
potential of A. Because the cardinal of FA∩FC
n
is O(n), the nodes on the
boundary FC
n
add at most O(n)αmore to the potential U
∂
n
(A)thantothe
potential U
⊘
n
(A)ofA, when all nodes of the boundary ∂C
n
of C
n
are in state 0.
If A°hasO(n
2
)nodesandαis small, the ratio Uϕ
nAðÞ
U@
nAðÞ !
n!þ1 1andAdoes not
contribute to the phase transition, because exp O nðÞðÞ
exp O n2
ðÞðÞ
!
n!þ1 0. It is the same for
A
C
,ifA°
C
has O(n
2
)nodes.
In the second case of sets B, we show that the total number of Bs and their
potential fail to perturbate the dominant role of the As whatever the phase of
the process. The second case corresponds to sets B, whose total number of
boundary nodes, each contributing of u
0
to the potential of B,isO(n
2
).
Consider that FB has inodes in Q
1
and jin Q
2
, where Q
1
is the subset of
C
n
made of the origin of ℤ
2
and of all points of ℤ
2
having even coordinates,
and Q
2
represents the subset of C
n
containing all points of ℤ
2
having odd
coordinates. Then, the total number of such Bs is less than
Xn2
2
i¼0Ci
n2
2Xn2
22i
j¼0Cj
n2
2
;(45)
and their contribution P(C
n
) to the normalisation constant Z
n
of the
Boltzmann measure on the set of subsets of C
n
is less than
Pn2
2
i¼0Ci
n2
2Pn2
22i
j¼0Cj
n2
2
exp iþjðÞαðÞ
Pn2
2
i¼0Ci
n2
2
exp iαðÞ1þexpðαðÞÞ
n2
22i
¼1þexpðαðÞÞ
n2
21þexp α2ðÞðÞ
n2
22þαðÞ
n2
21þ1þα
exp 2ðÞ

n2
2¼Bn
:(46)
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This upper bound B
n
of the Bs’contribution to Z
n
is less than the contribution of
all the subsets Dverifying Uϕ
nDðÞ
U@
nDðÞ !
n!þ1 1toZ
n
. Among these subsets, we find the
subsets Amentioned above, plus all the subsets Dfor which FD∩FC
n
=ø.Ifβis
small, the contribution of the Dsisgreaterthan:
Xn24n
j¼0ejβCn24n
j1þexp βðÞðÞ
n24n2þβðÞ
n24n¼bn:(47)
Then the upper bound of Eq. (46) being less than the lower bound of Eq. (47)
with Bn
bn!
n!þ1 0, there is no phase transition when αand βare small.
5.5. Assessing a public health policy against obesity
We simulate Eq. (6) or its Hopfield analogue Eq. (27) by considering 100
individuals in the empirical network 2. We assume that all individuals have a
tolerance equal to 1 and that most individuals are obese. We consider that a
public health policy of preventive and therapeutic education has three options:
●The first action plan consists in doing nothing and leaving the population
structure converge to its stable state.
●In the second action plan, after 40 iterations, the 33 most influential
individuals (one-third of the whole population, corresponding to the
smallest proportion allowing a 10% reduction in the total number of
obese) are educated and supposed to remain or become normal-weight
or overweight. Influence is measured by the centrality coefficient, which
depends on the adjacency matrix of the interaction graph of the network
(Bonacich, 1987; Newman, 2008).
●In the third action plan, the 33 individuals are chosen at random.
Figure 10 presents the simulations:
●In the first action plan, starting from 97% of obese, 100% of the indivi-
duals become obese at equilibrium.
●In the second action plan, starting from 97% of obese, about 90% of the
individuals are obese at equilibrium.
●In the third action plan, starting from 67% of obese, about 40% of the
individuals are obese at equilibrium.
The results depend on the initial proportion of obese, as well as on the choice
of the individuals to be educated: Figure 10 Bottom shows that the random
choice performs better than the choice based on centrality. This difference
results from the fact that centrality in a directed graph is calculated with
240 J.DEMONGEOTETAL.
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outgoing edges only, but a central individual can also receive ingoing edges
from obese individuals, leading him and his neighbours to become obese,
which is less likely if individuals are chosen at random. The model shows that
the prophylaxy of obesity must promote a preventive education on diet and
physical activity (Demongeot et al., 2015b,2016b) addressed to a sufficient
percentage of obese chosen at random.
6. Conclusion
In the model of Eq. (4), (5), and (6), or its Hopfield analogue of Eq. (26) of
contagion and diffusion, we have weaved together individual and population
levels to simulate the dynamic of obesity in a social network. We have
redefined contagion and diffusion at the individual level. We have simulated
the spread of obesity using a Boolean formalization of social networks, which
we have embedded within a differential system. These simulations fit for the
data collected in the French high school of Jœuf, which allows us to validate
Figure 10. Top left-hand side (a): 97% of the individuals are initially obese with tolerance =
1. In absence of policy for reducing obesity, all individuals become obese. Top middle (b):
all the individuals have a tolerance 1, 67% are initially obese, and the 33 most central
individuals are educated to remain or to become normal-weight or overweight, leading to
a stable state of around 90% of obese. Top right-hand side (c): all the individuals have a
tolerance 1; 67% are initially obese and 33 individuals are randomly chosen to be
educated. Then, the rate of obese falls to 40%. Bottom: sensitivity of the final percentage
of normal and overweight to the percentage of obese to be educated chosen as the most
central individuals (blue) and as random (red).
MATHEMATICAL POPULATION STUDIES 241
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the hypothesis of homophily. The model is the single one, to our knowledge,
to set health strategies while taking account of caregivers to obese and assess
a public health prophylaxy of obesity.
Acknowledgments
We acknowledge the Investissements d’avenir project Visual Home Presence inter@ctive and
the Maghrebi project “Systèmes complexes et ingénierie médicale.”We thank the reviewers
for their helpful comments. We dedicate this article to the French poet Eugène Guillevic,
born in 1907 and died in 1997, who gave a poetic definition of Poincaré’s stable focus (which
we have used in Section 5.2) as a spiral centre (Guillevic, 1967): Je sais qu’amenuisant,
/Durant mon aventure, /L’espace que j’enclave, /Je sais que tournoyant, /Autour de quelque
chose, /Qui est moi-même et ne l’est pas, /Je finirai par être, /Ce point auquel je tends: /Vrai
moi-même, le centre, /Et qui n’est pas. (English translation by Taylor (2003): I know, reducing
/During my adventure /The space I’m closing /I know, revolving /Around something /That is
both myself and is not,/I’ll wind up being /That point I’m aiming for:/A genuine me, the center
/And yet which is not one).
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