A mathematical model of anorexia and bulimia · 2020-03-06 · bulimic; anorexia class, A, in which an individual has the symptoms of anorexia; bulimia class, B, in which an individual

Research Article

Received 6 July 2013 Published online 10 September 2014 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.3270MOS subject classification: 34A34; 34D05; 34D20; 34D45; 37E25; 91E99; 93D05

A mathematical model of anorexia and bulimia

Carla Ciarciàa, Paolo Falsaperlaa, Andrea Giacobbeb*† andGiuseppe Mulonea

Communicated by E. Venturino

In this paper, we propose a mathematical model to study the dynamics of anorexic and bulimic populations. The modelproposed takes into account, among other things, the effects of peers’ influence, media influence, and education.We prove the existence of three possible equilibria that without media influences are disease-free, bulimic-endemic, andendemic. Neglecting media and education effects, we investigate the stability of such equilibria, and we prove that underthe influence of media, only one of such equilibria persists and becomes a global attractor. Which of the three equilibriabecomes global attractor depends on the other parameters. Copyright © 2014 John Wiley & Sons, Ltd.

Keywords: anorexia; bulimia; media influence; epidemic models; reproduction number; Lyapunov function

1. Introduction

The prevalence of eating disorders has increased over the last 50 years, and they have, recently, had a major impact on the physical andmental health of young women. Anorexia and bulimia are related to eating disorders. Both of these disorders revolve around the fearof obesity or obsessive desire to remain thin, and the biological necessity of consuming food.

In the USA, where statistics are generally complete and easy to access, 8 million people (90% of which are women, for this reasonstudies on eating disorders frequently look at women) suffer from eating disorders. Anorexia is suffered by 0.5% of women, 2–3% ofwomen suffer of bulimia [1]. Statistics reveal that the situation is really alarming: in some EU countries, 0.93% of woman older than 18suffer from anorexia. In particular, in Italy eating disorders involve 3.3% of woman and man older than 18 (see [2]). To these numbers,however, we should add another 8% of individuals who do not show all the features, which are essential for the diagnosis of anorexiaor bulimia, but have sub-clinical forms of the diseases.

These disorders are very serious: anorexia nervosa is the third most common chronic illness in the USA [3]. In Australia, eating dis-orders are the seventh major cause of mental disorders, and treatment for anorexia nervosa represents the second highest cost to theprivate hospital field [4].

Although eating disorders are prevalent in western countries, recent studies have shown that the incidence of anorexia has risensharply in Asian countries such as China and westernization is one of the causes of the development of eating disorders in Chinesepopulation [5].

1.1. Harmful psychological influences and prevention

In Ref. [6], the authors investigate the meaning of body image and the role it plays during the adolescence. Body image is the inter-nal representation of one’s own outer appearance, which reflects physical and perceptual dimensions. In the age range 10–15 years,20–50% of girls in the USA say that they feel too fat [7] and 20–40% of girls feel overweight [8]. An important study has shownthat 40% of adolescent girls believed that they were overweight, even though most of these girls fell in the normal weightrange [9].

a Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italyb Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy* Correspondence to: Andrea Giacobbe, Università degli Studi di Padova, Dipartimento di Matematica, Via Trieste 63, 35121 Padova, Italy.† E-mail: [email protected]

Copyright © 2014 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2015, 38 2937–2952

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The family acts as a primary socialization agent by transmitting certain messages to adolescents, often differently according togender [10]. Peers also are important in shaping body image and eating patterns. Girls who compare their appearance with that oftheir female peers have a greater risk of body dissatisfaction [11–14].

Media’s effect on adolescent girls is strikingly strong [15, 16]. Studies from the USA, Britain, and New Zealand offer evidence thatincreased media use, especially the number of hours per day spent watching television, is associated with greater Body Mass Index andgreater risk of obesity among children and adolescents [17, 18].

Media propose also an unrealistic ideal to be thin. In particular, investigators have explored the hypothesis that an increasinglythin standard of female beauty has led to increases in weight and shape anxiety, dieting, and disordered eating in girls and women.Investigators from a range of disciplines (e.g., anthropology, communications, history, philosophy, and psychology) have used a varietyof methods to examine the relationship between media and how girls and women regard their bodies [19–22]. Important works arethose of the anthropologist Ann Becker [21, 22]. In her studies of Fijian girls’ self perception during the 3-year period in which westernmedia were introduced to Fiji, Becker observed that dieting and disordered eating appeared in adolescent girls for the first time everin Fijian culture. The influence of Western media in Fiji is particularly significant given that the thin ideal of beauty directly contradictstraditional Fijian norms. In another Australian qualitative study, girls associated the media’s portrayal of the thin ideal with pressure tobe thin [19].

Messages about body weight and appearance are now common also in the Internet. Although there are many sites that conveypositive health messages to young people, several web sites contain health-related information that can be harmful, portraying disor-dered eating in a positive light. A very deep and interesting analysis of pro-eating disorder web sites can be found in Ref. [23] wherethe authors describe different kind of messages to which users are exposed. These sites characterize anorexia and bulimia as a lifestylechoice, not a clinical disease [6, 24, 25].

Frequent magazine readers, usually adolescent girls, also are more likely to engage in anorexic and bulimic behaviors, such as takingappetite control or weight-loss pills. Research suggests that several factors contribute to harmful attitudes and behaviors, but exposureand desire to resemble media ideals are significant factors that must be taken into consideration [26, 27].

For many patients suffering from severe anorexia nervosa, hospitalization does not lead to full remission, because, typically, residualpsychopathological features persist after weight-recovery [28]. In fact, some individuals achieve complete recovery while others areravaged by a chronic disorder, and some die from it. Predicting course and outcome of anorexia nervosa is complicated by the intrinsiccomplexity of the disorder [29, 30].

Eating disorders research has moved toward attempted prevention. To prevent eating disorders, one needs to first understand whatcauses them and then to institute programs in order to mitigate those causes or to teach individuals how to deal with them. There aremany educational campaigns to prevent eating disorders promoted by schools, colleges, social institutions, and so on [31]. Accordingto the National Institute of Mental Health, it is important to increase the awareness that eating disorders are a public health problemand that prevention efforts are warranted [32–34], especially prevention at school [35]. Furthermore, researches revealed significantreductions in disordered eating patterns and disturbed attitudes about eating and body shape, as well as significant increases in healthyeating patterns after a prevention program also in a high-risk school setting [36, 37].

1.2. Mathematical models

Mathematical epidemiology has grown exponentially starting in the middle of the 20th century, and a great variety of models havebeen formulated, mathematically analyzed, and applied to infectious diseases. In the recent years, models have been formulated tocontrol the 2002–2003 epidemics of SARS [38, 39], the H1N1 influenza of 2009 [40–43], and to predict negative habits and socialbehaviors [44–49].

The time evolution of anorexia and bulimia has been also analyzed in the context of epidemiological models (see for example[50–52]). In this work, we focus on the spread of anorexia and bulimia nervosa and investigate a mathematical model in which anorexiaand bulimia depend not only on peer pressure (related to parameters ˇ1,ˇ2) but also on the influence of media (related to param-eters m1, m2). This last factor has a strong influence on the evolution of the system. As it will become apparent from the dynamicalequations (1), the influence of media causes the disease-free equilibrium to disappear. Recovery from these pathological conditions canbe obtained through pharmacological therapy with antidepressants and with cognitive-behavioral therapy, which fosters the devel-opment of healthy body images in order to prevent re-sensitization (i.e., to become susceptible once again). We model the effects oftreatment using the parameters �1, �2 and of (re)sensitization using the parameter � [53].

We also study the positive effect of a parameter related to education, which we call � . In this model, we consider only the possibilitythat anorexic individual can become bulimic because the rate of bulimic individuals that become anorexic can be disregarded in a firstapproximation. The main scope of this research is to consider the positive effects of education and the negative effects of some media,to compute their influence on the reproduction numbers and the equilibria, and finally to investigate strategies to mitigate the effectsof the disorders on population acting on such parameters.

In Section 2, we describe and formalize our model, which we denote SABR, because the compartments are susceptible, anorexic,bulimic, and recovered. After a proof of positive invariance of the admissible region and of the existence of three equilibria inSection 3, we consider at first, in Section 4, the case in which the influence of media and education are neglected. In such case,we analyze existence and spectral stability of the equilibria, global stability of the disease-free equilibrium with a Lyapunov func-tion, basic reproduction number, and its sensitivity with respect to the parameters. In Section 5, we finally discuss, partly analyticallyand partly numerically, the effect of education and media. We numerically prove the existence and the global stability of a uniqueendemic equilibrium.


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2. The SABR model

Our model is inspired by an article of Gonzalez et al. [54], in which a general model is suggested for anorexia and bulimia considered asepidemics. In that article, the authors restrict their attention to the spread of bulimia by dividing the infected individuals in two classesand analyzing the existence of endemic equilibria. That article concludes the analysis fixing the parameters according to previous med-ical literature, and numerically investigating the evolution of simple and advanced bulimic depending on the net infective force. Weextend this investigation to a model that describes both infective classes: anorexia and bulimia but considering only one group of indi-viduals for each class. Our model includes several alternative routes of infection/recovery: peer pressure, media effect, and education.In particular, we divide the population into four classes: susceptible class, S, in which individuals are at risk of becoming anorexic orbulimic; anorexia class, A, in which an individual has the symptoms of anorexia; bulimia class, B, in which an individual has the symp-toms of bulimia; and recovered/educated class, R, in which individuals have been taught healthy eating behaviors and body images.We are able to perform the investigation in a rigorous mathematical setting almost up to the general case, and we resort to a numericalinvestigation only at the very end, to prove with certainty the existence of a unique endemic equilibrium.

Let S, A, B, R denote, respectively, the number of susceptible individuals, the number of anorexic individuals, the number of bulimicindividuals, and the number of recovered/educated individuals. The at-risk population S can develop either anorexia A or bulimiaB because of contact with peers or the influence of media. Once anorexic, an individual may become bulimic. An anorexic orbulimic individual can recover from this condition, and move to the recovered class R. Once recovered, an individual may becomeagain susceptible.

According to the Introduction, we consider also the case in which a susceptible becomes not sensitive to negative peer pressuresand media influences, thanks to an education campaign.

The model is described by Figure 1 assuming that the parameters appearing near the arrows are multiplied by the class from whichthe arrows go out, as proposed by Hethcote [55]. This model is more appropriate to describe a homogeneous population (for instanceyoung women in the age range 12–25 years), because such part of the population is primarily at risk. In fact, women are more suscep-tible than men, and they enter the susceptible group as they enroll in Junior High School and begin frequenting other adolescents,while they leave the group by finding a job or creating a family of their own.

The parameters of the model, all non-negative constants, are

� m1: rate of individuals becoming anorexic because of media influences per unit time;� m2: rate of individuals becoming bulimic because of media influences per unit time;� ˇ1: anorexic individuals’ peer-pressure contact rate per unit time;� ˇ2: bulimic individuals’ peer-pressure contact rate per unit time;� ˛: rate of anorexic individuals that become bulimic per unit time;� �1: rate of anorexic individuals that recover for medicine or because of social campaigns per unit time;� �2: rate of bulimic individuals that recover for medicine or because of social campaigns per unit time;

Figure 1. The SABR model that describes the spread of eating disorders in a community of susceptible (S), anorexic (A), bulimic (B), and recovered people (R).Arrows indicate the direction of movement into or out of a group.


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� � : education rate per unit time;� �: entry and exit rates of the general population per unit time;� �: sensitization rate.

Observe that the class of recovered population R contains the individuals that have healthy body images and contains individualsthat have never had eating disorders but are immune through education or because of a strong personality, and those who have hadthem but have been treated. The (re)sensitization rate � we use in our model is an average of the sensitization rates of the two families.

We assume that the rate at which anorexia and bulimia spread depends on how often susceptible individuals meet people with eat-ing disorders and how successful those encounters are in transmitting eating disorder habits, and how persuasive media are. Thesesocial factors are embedded in the recruitment rate as we noted previously. The number of individuals who develop eating disordersdepends on the relative sizes of the healthy and ill population. The probabilities of meeting an anorexic or a bulimic individual is pro-portional to the fraction of the two groups A=P and B=P in the total population P D SC AC BC R, and such contacts can drive someindividuals to the corresponding eating disorder. This fact (perhaps surprising given the visible devastation suffered by many anorexicindividuals) is well documented in the literature [56–59]. In the model that we consider, the possibility that a bulimic individual becomesanorexic is disregarded because, according to the American Psychiatric Association, half of anorexic patients do develop bulimia,while only a few bulimic patients develop anorexia. This fact is in accordance with the introduction of [60] and with the mathematicalmodel of [54].

This model of spread of anorexia and bulimia can be cast mathematically as a set of the following four nonlinear ordinary differentialequations that describe the changes in the populations S, A, B, and R over time:

8̂̂ˆ̂̂̂̂ˆ̂̂̂̂<ˆ̂̂̂̂̂ˆ̂̂̂̂:̂

dS

dtD �P � .�C � Cm1 Cm2/S � .ˇ1AC ˇ2B/

S

PC �R

dA

dtD m1SC ˇ1A

S

P� .�C ˛ C �1/A

dB

dtD m2SC ˇ2B

S

PC ˛A � .�C �2/B

dR

dtD �SC �1AC �2B � .�C �/R .

(1)

As discussed previously, we assume that the population under study is part of a larger population at demographic equilibrium, sothat we can take it to be constant. It is hence natural to normalize the quantities by introducing the new variables to be constant, withequal entry and exit rates �,

S D s � P A D a � P B D b � P R D r � P ,

obtaining the normalized model

8̂̂ˆ̂̂̂̂̂ˆ̂̂̂<ˆ̂̂̂̂̂ˆ̂̂̂̂̂:

ds

dtD � � .�C � Cm1 Cm2/ s � .ˇ1 aC ˇ2 b/ sC � r

da

dtD m1 sC ˇ1 a s � .�C ˛ C �1/ a

db

dtD m2 sC ˇ2 b sC ˛ a � .�C �2/ b

dr

dtD � sC �1 aC �2 b � .�C �/ r

(2)

subject to the constraint

1 D sC aC bC r. (3)


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We hence can use the integral of motion (3) and reduce the normalized system (2) to a system consisting of the three differentialequations 8̂̂̂

ˆ̂̂̂<ˆ̂̂̂̂̂:̂

da

dtD �ˇ1 a2 � ˇ1 a b � ˇ1 a rC .ˇ1 �m1 � � � ˛ � �1/ a �m1 b �m1 rCm1

db

dtD �ˇ2 b2 � ˇ2 a b � ˇ2 b rC .˛ �m2/ aC .ˇ2 �m2 � � � �2/ b �m2 rCm2

dr

dtD .�1 � �/ aC .�2 � �/ b � .� C �C �/ rC � .

(4)

We observe that in this general case, the disease-free equilibrium, that is, the case in which the whole population belongs to the sus-ceptibles or the removed, does not exist. In fact, if all the parameters are positive constants, to the choice a D b D 0 correspond apositive time derivative of the solutions a.t/, b.t/.

The analysis of stability of equilibria for this system is complicated. In the following sections, we consider some particular cases wherethis investigation becomes possible, and we use the results to draw conclusions for the general case. We start by considering m1, m2

and � equal to zero, we then consider the case � > 0 with m1, m2 D 0, and finally, we investigate the effect of media (m1, m2 > 0) onthe equilibria.

3. General properties of the model

3.1. Positive invariance of the unit tetrahedron

Representing percentages of a population, the three quantities a, b, r must be positive and have sum less than 1, that is, must belongto the tetrahedron

T D f.a, b, r/ j aC bC r � 1, a , b , r > 0g.

In this section, we analyze the positive invariance of such tetrahedron, that is, we show that any solutions starting inside that regioncan never leave it.

Positive invariance is equivalent to the fact that the vector field X whose associated O.D.E. is the system of equations (4) is alwaysentering the faces of the tetrahedron, that is, its scalar product with the inner normal vector of the boundary of T is always positive.The tetrahedron is composed of four faces:

� The face of T lying in the b, r-plane has inner normal .1, 0, 0/, and its scalar product with X is m1.1�b� r/. This function is positiveon that face, in which precisely bC r < 1 (and b, r > 0).

� The face of T lying in the a, r-plane has inner normal vector .0, 1, 0/, and its scalar product with X is a˛�a m2�m2 rCm2. Lettingh D ˛=m2, the equation becomes .1 � h/aC r < 1, and this equation is satisfied in a set that includes the face of T .

� The face of T lying in the a, b-plane has inner normal vector .0, 0, 1/, and its scalar product with X is a .�1 � �/C b .�2 � �/C � .Denoting h D �1=� and k D �2� , the equation becomes a .1�h/Cb .1�k/ < 1, and this equation is satisfied in a set that includesthe face of T .

� The face of T inside the first octant has equations aCbC r D 1 and inner normal n D .�1,�1,�1/. The scalar product n �X equals�C � r, which is positive on the face.

This proves the positive invariance of T . We will prove that this system always has three equilibria, but only some of them belong toT , and hence have meaning in this model. In the sequel, we will say that an equilibrium exists or that it is admissible if it belongs to T .The main goal of our treatment is to determine the existence and stability of admissible equilibria.

3.2. The equilibria

In this section, we prove that the system always admits three equilibrium solutions. To be meaningful, an equilibrium must havecoordinates a, b, r, which are positive and such that aC bC r � 1.

From the first and third components of the vector field at the equilibrium, one has that, posing � D �C �,

1 � r.a, b/ Dˇ1a2 C ˇ1abC .m1 C �C ˛ C �1/aCm1b

ˇ1aCm1(5)

r.a, b/ Da �1 C b �2 C � .1 � a � b/

�C �(6)


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(in our model, we assume �C � 6D 0.) These two equations imply, respectively, that r.a, b/ < 1 when a, b are positive and r.a, b/ > 0when a, b have a sum less than 1, the equilibrium point .a, b, r.a, b// is always admissible whenever a, b > 0 and aC b < 1.

Substituting (6) in the first two components of the vector field and equating to zero, one obtains the two equations:

.a ˇ1 Cm1/.c1 C a.�1 C �/C b.�2 C �// D c2 (7)

��bˇ2 �m2 �

˛.�C �/

�1 C �

�.c3 C a.�1 C �/C b.�2 C �// D c4 (8)

with

c1 D.˛ C �1 C �/.�C �/

ˇ1� �, c2 D m1

.˛ C �1 C �/.�C �/

ˇ1, c3 D

˛.�2 C �/.�C �/

ˇ2.�1 C �/C.�C �/.�2 C �/

ˇ2� �

and

c4 D�C �

ˇ2

�m2.�2 C �/ �

˛2.�2 C �/.�C �/

.�1 C �/2�˛.�� C �.�ˇ2 C �2 C � �m2/C �2.� �m2//

�1 C �

�

It is clear that, unless ˇ1 D 0 or ˇ2 D 0, such equations are those of two hyperbolas, and we are in the case in which two asymptotesare parallel; hence, the two hyperbola intersect only in three points (see the Appendix). When either ˇ1 or ˇ2 is zero, then some of thealgebraic passages need a little more care, and one can see that one of the two hyperbolas degenerates to a line, and the intersectionthen consists of two points. If both ˇ1 D ˇ2 D 0, then the intersection is a unique point. We note that in our model, we assume alwaysˇ1 > 0, ˇ2 > 0, so the system presents three equilibria.

The expression of the equilibria in the generic case is too cumbersome to write explicitly, and it is difficult to discuss existence andstability. We begin our investigation with the case in which the effect of media and education are absent (i.e., m1 D m2 D � D 0). Wethen discuss partly analytically and partly numerically what happens when these parameters move away from zero.

4. The simplified case: m1 D m2 D � D 0

Disregarding media and education coefficients m1, m2, � , system (4) becomes

8̂̂ˆ̂̂̂̂<ˆ̂̂̂̂ˆ̂:

da

dtD �ˇ1 a2 � ˇ1 a b � ˇ1 a r C .ˇ1 � � � ˛ � �1/ a

db

dtD �ˇ2 b2 � ˇ2 a b � ˇ2 b r C ˛ aC .ˇ2 � � � �2/ b

dr

dtD �1 aC �2 b � � r.

(9)

We see immediately that the disease-free equilibrium E0 D .0, 0, 0/ is a solution of (9).To discuss the local stability of E0, we consider the Jacobian matrix J0 associated with system (9) in E0. A simple computation gives

J0 D

26664ˇ1

Ra�1Ra

0 0

˛ ˇ2Rb�1

Rb0

�1 �2 ��

37775 ,

where we introduced the quantities

Ra Dˇ1

�C ˛ C �1, Rb D

ˇ2

�C �2.

The eigenvalues of matrix J0, because it is lower triangular, are given by the diagonal elements, and then local stability of E0 is ensuredby the conditions Ra < 1, Rb < 1. The quantity R0 defined by

R0 D maxfRa, Rbg


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guarantees that the disease-free equilibrium is linearly stable for R0 < 1; instead, if R0 > 1, the pathological behaviors will spread inthe susceptible population. So R0 is the basic reproduction number. We compared this result with the computation of R0 based on thenext generation operator approach introduced by Diekmann et al. [52, 61] (a number of salient examples of this method are in [62–64]).We use the unreduced system (2), for which the disease-free equilibrium has s D 1 and a D b D r D 0. The disease compartments arein this case a and b, so that, using the notations of the previous references, we have

F D

0@ˇ1a s

ˇ2b s

1A V D

0@ .�C ˛ C �1/a

�˛aC .�C �2/b

1A .

The Jacobian matrices of F and V on the disease-free equilibrium are

F. 1, 0, 0, 0/ D

�ˇ1 00 ˇ2

�V. 1, 0, 0, 0/ D

��C ˛ C �1 0�˛ �C �2

�,

and so the next generation operator is

FV�1 D

0@ Ra 0

˛ˇ1

RaRb Rb

1A

whose spectral radius is

R0 D maxfRa, Rbg

which coincides with our previous result. A simple analysis shows that R0 is most sensitive to changes in the value of ˇ1 or ˇ2.

4.1. Global stability of E0

We want to prove global stability of the equilibrium E0 using the theory of Lyapunov functions [65]. Let us consider the two-parametricfamily of functions

V.a, b, r/ D aC h bC k r

with h, k strictly positive reals. Note that V.E0/ D 0 and V > 0 for any .a, b, r/ 6D E0 in the positive octant. To prove global stability of theE0 equilibrium, we compute the orbital derivative of V , which is

PV D �ˇ1.aC bC r/aC .ˇ1 � � � ˛ � �1/a � hˇ2.bC aC r/bC h˛ aC h .ˇ2 � � � �2/ b C k.�1aC �2b/ � k � r.

From a linear stability analysis, we know that E0 is locally stable if

Ra < 1, Rb < 1, that is, ˇ1 � � � ˛ � �1 < 0, ˇ2 � � � �2 < 0, (10)

but these conditions do not guarantee in general PV < 0. We investigate then the effect of different values of h, k. By choosing h D k D 1,one obtains that

PV < .ˇ1 � �/ a C .ˇ2 � �/ b,

and global stability follows when ˇ1, ˇ2 < �. A stricter condition can be obtained, observing that

PV < .ˇ1 � � � ˛ � �1/a C h˛ aC h .ˇ2 � � � �2/ b C k.�1aC �2b/ D

D .ˇ1 � � � ˛.1 � h/ � �1.1 � k//aC h

�ˇ2 � � �

�1 �

k

h

��2

�b,

and PV is globally negative for

ˇ1 � � � ˛.1 � h/ � �1.1 � k/ < 0, ˇ2 � � �

�1 �

k

h

��2 < 0.

Such conditions can be made arbitrarily close to the linear instability conditions (10) if we choose 0 < k � h � 1. We have henceproved that when the equilibrium E0 is spectrally stable, then it also is globally stable in the positive octant.


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4.2. The endemic equilibria

In this section, we calculate the endemic equilibria of the model and introduce the quantities

�1 D ˇ1Ra � 1

Ra�2 D ˇ2

Rb � 1

Rb

to simplify our computation, which imply immediately

Ra > 1 , �1 > 0 Rb > 1 , �2 > 0

So, we consider again system (9), and find the further two equilibria:

� E1 D�

0, ��2ˇ2.�C�2/

, �2�2ˇ2.�C�2/

�the anorexia-free endemic equilibrium, where only bulimia is endemic, which we call for the sake

of shortness bulimic-endemic equilibrium; and

� E2 D��1 �1ˇ1 �2

, �˛�1�2

, �1�1 �1ˇ1 �2

C �2˛�1�2

�, the endemic equilibrium;

where

�1 D ˇ1ˇ2Ra � Rb

RaRbD �1ˇ2 � �2ˇ1

�2 D �1.�C �1/C ˛ˇ1.�C �2/.

RemarkWhen the coordinates of an equilibrium are positive, then their sum is less than or equal to one. In particular, the sums of the coordinatesof E1, E2 are �2=ˇ2 and �1=ˇ1, respectively.

Let us now investigate existence and stability of E1 and E2 relative to the stability conditions of E0.The expression of E1 shows that this equilibrium exists only when Rb > 1 (i.e., �2 > 0) and hence when E0 becomes unstable. The

stability of E1 can be determined by evaluating the Jacobian matrix J1 in E1

J1 D

266664

�1ˇ2

0 0

˛ � ��2�C�2

� ��2�C�2

� ��2�C�2

�1 �2 ��

377775 .

One eigenvalue is �1=ˇ2, and the sum and product of the other two eigenvalues are, respectively,

�� 2

�C �2, ��2 .

Because E1 exists for �2 > 0, these eigenvalues are negative and stability of E1 depends entirely on the sign of �1. Then E1 is spectrallystable only if �1 < 0 and then Ra < Rb.

The expression of E2 shows that this equilibrium exists if �1 > 0 (i.e., Ra > Rb) and �1=�2 > 0. Because �1 > 0 implies �2 > 0, theseconditions turn out to be Ra > Rb and Ra > 1.

The Jacobian matrix J2 associated with the equilibrium E2 is

0BBBB@��1�1

�2��1�1

�2��1�1

�2

˛ � ˛�ˇ2�1�2

� �1ˇ1� ˛�ˇ2�1

�2�˛�ˇ2�1

�2

�1 �2 ��

1CCCCA

The characteristic polynomial of J2 is

�3 C a2�2 C a1�C a0 D 0


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where

a2 D �tr.A/ D��1�1

�2C�1

ˇ1C˛ �ˇ2�1

�2C �

a1 D��1ˇ1�1�1 C �1ˇ1�2˛ ˇ2 C �1�2 C ˛ �ˇ2�1ˇ1 C ��1�1ˇ1 C �1�1˛ ˇ1 C �1�1

2�

ˇ1�2

a0 D �det.A/ D��1�1

ˇ1.

We notice that these coefficients are all positive. To prove the local stability of E2 by the Routh–Hurwitz criterion, we should alsoprove that

a2 a1 � a0 (11)

is positive. This condition is difficult to be proved analytically because of the many parameters. However, numerical sampling of expres-sion (11) in the space of parameters and a numerical minimization of (11) show that when E2 exists, it is also locally stable. We cansummarize these results in Table I.

It is instructive to analyze the existence of the equilibrium E1 in the plane ˇ1,ˇ2 when the other parameters are fixed. The geometryof such region is always qualitatively the same: the half-plane Rb > 1 (Figure 2).

Table I. The scheme of equilibrium points and their stability for m1 D m2 D � D 0.E0 E1 E2

Ra < 1, Rb < 1 Stable Does not exist Does not exist

Ra > 1; Rb < 1 Unstable Does not exist Stablea

Ra < 1; Rb > 1 Unstable Stable Does not exist

Ra > 1; Rb > 1, Ra < Rb Unstable Stable Does not exist

Ra > 1; Rb > 1, Ra > Rb Unstable Unstable Stablea

aThe stability of E2 is proved only numerically.

Figure 2. The equilibria E1 and E2 are stable for choices of parameters in the dark and light shaded regions, respectively. The regions are plotted in ˇ1,ˇ2-space,and they are qualitatively the same for every choice of the other parameters ˛,�1,�2,�, �. In this particular case, we have chosen ˛ D 0.3,�1 D 0.1,�2 D0.3,� D 0.01, � D 0.1. The disease-free equilibrium E0 is stable only when the parameters belong to the unshaded region, that is when Ra < 1 and Rb < 1.


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5. Case with influences of education and media

5.1. Case with � > 0 and m1 D m2 D 0

Let us now consider the effect of positive values of the education coefficient � . System (4) becomes

8̂̂̂ˆ̂̂̂<ˆ̂̂̂̂̂:̂

da

dtD �ˇ1 a2 � ˇ1 a b � ˇ1 a r C .ˇ1 � � � ˛ � �1/ a

db

dtD �ˇ2 b2 � ˇ2 a b � ˇ2 b r C ˛ aC .ˇ2 � � � �2/ b

dr

dtD �1 aC �2 b � � r C � .1 � a � b � r/.

(12)

In this case, the disease-free equilibrium is

E00 D

�0, 0,

�

� C �

�,

which is always admissible. Moreover, we note that in this state, the number of susceptibles is s D �=.� C �/.To discuss the stability of E00, we proceed as in Section 4.The Jacobian matrix J00 associated with system (12) at E00 is

J00 D

264

�01 0 0

˛ �02 0

�1 � � �2 � � ��

375

with

�01 D ˇ1R0a � 1

Ra, �02 D ˇ2

R0b � 1

Rb.

In these expressions, we introduce the new reproduction numbers

R0a D Ra�

� C �, R0b D Rb

�

� C �,

which include the fraction �=.� C �/ of the population susceptible to eating disorders in the disease-free state when an educationcampaign is considered. The eigenvalues of J00 are �01,�02 and�.� C �/, so the stability is guaranteed by the conditions R0a < 1, R0b < 1.We checked these results with the next generation operator approach, obtaining

FV�1 D

"R0a 0

˛ˇ1

R0aRb R0b

#

So, R00 D max�

R0a, R0b�

is the control reproductive number.It is straightforward that �01,�02, R0a, R0b are strictly decreasing functions of � . It follows that, as expected, � has a stabilizing effect on thedisease-free equilibrium.Also in this case, we have a bulimic-endemic equilibrium and an endemic equilibrium, but their explicit expressions and the study oftheir stability are mathematically cumbersome so we will not report them here.

5.2. General case

In this section, we describe what happens when the parameters m1, m2 become positive. It is possible to show partly analytically andpartly numerically that in this general case, there is always only one endemic equilibrium in the unit tetrahedron.

An increase in m1, m2 will increase the percentage of anorexic and bulimic individual, but which of the three possible equilibriaE00, E01, E02 described in the case without the influence of media will become the endemic equilibrium depends on the other parameters.

When m1 D 0 and m2 > 0, we denote by E000 , E001 , E002 the prolongation of the equilibria E00, E01, E02. It can be proved that E000 , E001 arebulimic-endemic but remain anorexic-free. Their analytic expression is


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Figure 3. The evolution of the b-component of the equilibria E000 , E001 as m2 becomes positive while m1 D 0. The left panel corresponds to a choice of parametersfor which E01 does not exist when m2 D 0 (that is,�2 < 0), and shows that the disease-free equilibrium becomes bulimic-endemic. The right panel corresponds toa choice of parameters for which E01 does exist when m2 D 0 (that is �2 > 0) and shows that the disease-free equilibrium exits from the admissible region whileE001 remains admissible.

�0,�

1

2ˇ2.�2 C �/

�G˙

pG2 C 4ˇ2m2�.�2 C �/

�,�

�(13)

where G D .�2 C �/� C .�2 C �/m2 � �2� and r is not explicitly written. Which of the two expressions (with plus or minus) is theprolongation of E00 and which of E01 is possible to say only once the parameters are fixed, and depends on the sign of G. From thisexpression, one can analytically prove that

Proposition 1As soon as m2 is increased from zero, then only one of the two anorexic-free equilibria (the disease-free E00 and the bulimic-endemic E01)will be in the unit tetrahedron, while the other will move out of the unit tetrahedron T . So there are two possible cases:

C0 The bulimic-endemic equilibrium E01 does not belong to the tetrahedron when m2 D 0; then also when m2 > 0, its prolongationE001 does not belong to the tetrahedron while E000 becomes bulimic-endemic.

C1 The bulimic-endemic equilibrium E01 does belong to the tetrahedron when m2 D 0; then also when m2 > 0, the equilibrium E001does belong to the tetrahedron and is anorexic-free and bulimic-endemic, while E000 moves out of the unit tetrahedron.

From now on, we call this bulimic-endemic and anorexic-free equilibrium E0001 (the subscript 01 indicates that the prolongation ofeither E00 or E01 plays the role of such equilibrium, and we cannot know a-priori which of the two will be). The two possible eventsdescribed in the aforementioned proposition are summarized in Figure 3.

As we discussed previously, when m1 D 0, there still exist two anorexic-free equilibria (i.e., with a D 0) that we denote E000 , E001 . Thevalue of their b-component is written in formula (13).

Regardless the sign of G, one of the two components becomes negative while the other becomes bulimic-endemic (i.e., with bpositive) and anorexic-free (i.e., with a D 0). Which of the two depends on the sign of G. There are hence two possibilities described inthe previous result. The plots of the two possible scenarios is depicted, for two different choices of all the parameters except m2 (andwith m1 fixed to zero) in Figure 3 left for the case C0 and in Figure 3 right for the case C1.

When m2 > 0 and also m1 is increased from zero, the coordinates of the equilibria do not have simple analytical expression. They arethe roots of a cubic polynomial in a whose coefficients depend on the parameters and hence can be obtained using Cardano’s formula.Not only the investigation of their positivity is extremely difficult but also to decide which of the three expressions tend to E000 , E001 , E002 ,respectively, when m1 tends to zero is complicated. We outline the evolution of the equilibria as m1 grows away from zero by resortingto the numerical analysis plotted in Figure 4. Also in this case, there are two possibilities.

Fact 2If m1 6D 0, m2 6D 0, we denote by E0000 , E0001 , E0002 the three equilibria that tend, respectively, to E000 , E001 , E002 as m1 tends to zero. Only oneof such solutions lies in the interior of the unit tetrahedron T , giving a system with precisely one endemic equilibrium. There are twopossibilities:

D01 The endemic equilibrium E002 does not belong to the tetrahedron when m1 D 0; then also when m1 > 0, the equilibrium E0002 doesnot belong to the tetrahedron while E0001 becomes endemic (i.e., either E0000 or E0001 moves in the interior of the unit tetrahedron).

D2 The endemic equilibrium E002 does belong to the the unit tetrahedron when m1 D 0; then also when m1 > 0, the equilibriumE0002 belongs to the tetrahedron and remains endemic. In this case, E0000 and E0001 move out of the unit tetrahedron (one of themalready did not belong to such tetrahedron already when m1 D 0).


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Figure 4. The evolution of the a-component of the equilibria E0000 , E0001 , E0002 as m1 becomes positive. The solid lines represent equilibria whose b-component ispositive; the dotted lines to equilibria whose b-component is negative. The left panel is associated with a choice of parameters for which E002 does not exist, theright panel to a choice for which E002 does exist when m1 is set to zero.

Figure 5. Orbits of system (12) starting from a regular grid of points inside the unit tetrahedron and converging to the equilibrium point indicated by a smallcircle. Dashed lines show the projection of the equilibrium point on the coordinate planes. In the left panel, we show a global endemic equilibrium E02, with� D 0.05. In the right panel, we change only � to 0.1 obtaining a global disease-free equilibrium.

In this case, the two possible scenarios can be proven only numerically. In the left panel of Figure 4, we plot the case D01 under thehypothesis that C0 is verified. In the a-axis, there are only solutions with a D 0. Such solutions are the two E000 and E001 . Increasing m1

the two a-components of those solutions become positive, but one of them has negative b-component, while the other has positiveb-component and is hence admissible, that is, belongs to the unit tetrahedron.

In the right panel of Figure 4, we plot the case D2. In the a-axis, there are two solutions with a D 0�

E000 and E001�

and one with positivea�the endemic equilibrium E002

�. Increasing m1, the two a-components of E000 , E001 become negative, and these solutions are hence non-

admissible. One of them also has negative b-component, while the other has positive b-component. On the other hand, the equilibriumE0002 has a-component, which increases with m1, and remains an endemic solutions with higher percentage of anorexic individuals whenm1 becomes larger.

5.3. Numerical illustration

In this section, we examine numerically the competing effect of the education factor � and the media influence on the onset of anorexiam1 and bulimia m2. We consider the initial set of parameters ˇ1 D 0.4,ˇ2 D 0.3, m1 D m2 D 0,˛ D 0.05, �1 D 0.05, �2 D 0.2,� D0.05, � D � D 0. With this choice of parameters, we have Ra D 2.67 > Rb D 1.2 > 1, and, as expected, the only endemic equilibriumE2 D .0.16, 0.06, 0.40/ (Table I).

Introducing the education factor � D 0.05, the new reproductive numbers are R0a D 1.3, R0b D 0.6. The endemic equilibrium isE02 D .0.07, 0.02, 0.54/, which, by numerical evidence, is still globally stable (Figure 5 – left). What we see is that both the anorexic andbulimic populations have noticeably shrunk but they are still present.


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Figure 6. Orbits of system (4) starting from a regular grid of points inside the unit tetrahedron with a different choice of parameters. In the left panel, we show theeffect of media pressure on bulimic population (m2 D 0.05, � D 0.1). In the right panel, we show the effect of media on anorexic population (m1 D 0.05, � D 0.1).

A further increase of � to � D 0.1 has the effect of making R0a D 0.89 and R0b D 0.4 both less than 1. The numerically globally stableequilibrium is in this case the disease-free state E00 D .0, 0, 0.67/ (Figure 5 – right). The effect is exactly what we expected from a strongeducational campaign.

We introduce now the negative influences of media on bulimia, by setting m2 D 0.05. Note that we have still a strong educationaleffect from � D 0.1, but we know that a state with no bulimic individuals can no longer be an equilibrium, because now, db=dt 6D 0even for b D 0. The new bulimic-endemic equilibrium E001 D .0, 0.06, 0.71/ is shown in Figure 6 – left.

As expected, a worse effect derives from a promotion of anorexic behavior, modeled in our numerical computation by m1 D

0.05, m2 D 0. The new equilibrium is .0.13, 0.03, 0.65/. Even in this case, there is numerical evidence that such state is globally stable(Figure 6 – right).

6. Final comments

In this paper, we propose a mathematical model for the dynamics of anorexic and bulimic populations. The model proposed takesinto account, among other things, the effects of peers’ influence ˇ1,ˇ2, media influence m1, m2, and education � . As far as we know,this problem with both kind of disease compartments has not been yet investigated, and we are not modeling a specific study on awell-delimited population. For this reason, the values of many parameters are purely indicative.

We begin the analysis by ignoring the effects of media pressure and education, and we obtain conditions for global stability of thedisease-free equilibrium introducing two reproduction numbers Ra, Rb associated with the anorexic and bulimic populations. We showthat there exist at most two endemic equilibria: the purely bulimic one and the endemic, both of which can be stable under certainconditions.

We then consider the influence of an educational campaign. In this last case, we notice that the reproduction number rescales withthe coefficient �=.� C �/, which indicates the fraction of population susceptible to eating disorder in the disease-free state when aneducation campaign is considered.

We finally study the case in which media influence plays a role. In such case, only one of the equilibria becomes endemic and belongsto the admissible region, while the other two become non-admissible. The equilibrium that becomes admissible is that which wasattracting when m1, m2 D 0, and as the media parameters are increased, it moves into the endemic region with increasing percentagesof anorexic and bulimic individual, remaining an attractor. Naturally, the final scenario is radically different whether the equilibriumthat becomes the attractor was disease-free, bulimic-endemic, or endemic when m1 D m2 D 0: the percentages of ill population differgreatly. Despite the effects of mass media, models such as this one serve the practical purpose of deriving reproductive numbers, whichcan predict the possible effects of combating media pressure. In particular, if R00 > 1, then even eliminating mass media influence willnot be sufficient to end the disorders.

What are the reasonable values of some of the coefficients is a complicated question. Any conjecture should be tested with the helpof the medical community. The values of the ratesˇ1,ˇ2, and their related unit of time are highly sensitive to the particular environmentin which the recruited population lives. For instance, such rates can be profoundly higher when dealing with a high-risk environmentsuch as a ballet school [37]. We have not found explicit values of these parameters in literature.

We conclude observing that, for simplicity, in this paper, we have made the assumption that the exit rate is the same for every com-partment and it is equal to the entry rate. A more general and biologically more significant model would require different entry andexit rates. The problem could also be enriched by adding multi-group components to capture heterogeneity in the mixing, adding arisk-structure or an age-structure, adding to the incidence functions a saturation term such as ˇ1A=.P C ˛A C ˇB/, and dividing therecovered in treated and non-susceptible, with different sensitization rates. Most of these refinements can be captured by an averaging


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assumption on population groups, while others will be made in future works, and some require a mathematical analysis that exceedsour capacities. For this reason, we have settled on this model, which could be a good compromise and could be in line with the sys-tems that are being considered nowadays by a mathematical community (of course, a numerical investigation with properly chosenparameters can easily be made on more complicate systems).

Appendix: on hyperbola and their intersections

In this appendix, we recall classical results on conics. Hyperbolas and parabolas in xy-plane are associated with equations of the formLM D c, where c is a real number and L, M are linear monomials of the form ˛x C ˇy C � with ˛,ˇ, � also real numbers. When the twolines L, M are parallel, the conic is a parabola; when they are not, it is a hyperbola. In such case, the two lines are the asymptotes of thehyperbola, and when c D 0, the hyperbola degenerates to the two lines L D 0, M D 0.

Two hyperbola L1M1 D c1 and L2M2 D c2 intersect in four points, unless they have parallel asymptotes. If only two asymptotes areparallel, that is, the equations are L1M D c1 and L2.MCd/ D c2, then the intersection points are only three; if all asymptotes are parallelin couples, that is, the equations are LM D c1 and .LC d1/.MC d2/ D c2, then the intersection points are only two.

This fact can be easily shown either directly (when L, M are non-parallel, then L, M are a good set of coordinates) or, more explicitly,by resorting to a translation and a linear change of coordinates that put L1 in the x-axis and M1 in the y-axis. In such case, the equationsof the two hyperbola are

yx D c, .˛1x C ˇ1y C �1/.˛2x C ˇ2y C �2/ D c1.

If the first hyperbola is non-degenerate, then solutions have x 6D 0, and hence, one can substitute y D c=x in the second equation toobtain the fourth-order equation in x

.˛1x2 C ˇ1cC �1x/.˛2x2 C ˇ2cC �2x/ D c1x2,

which has four solutions. If two asymptotes are parallel, then the equations become yx D c1 and .˛x C ˇy C �/.y C d/ D c2. With thesubstitution y D c1=x in the second equation, one obtains .˛x2Cˇc1C�x/.c1Cdx/ D c2x2, which is a third-order equation with threesolutions. In the last case, the equations are yx D c1, .xCd1/.yCd2/ D c2, and the substitution y D c1=x yields .xCd1/.c1Cd2x/ D c2x,which is a second-order equation and has two solutions.

Acknowledgements

The model presented in this work has been proposed by prof. W. Wang in occasion of a visit to the University of Catania where he gavea PhD course. Some preliminary results have been presented at the 5th CICAM meeting. We are indebted to prof. W. Wang for helpfuladvice.

The authors thank two anonymous referees who kindly read a previous version of the paper very carefully and drew our attention toseveral misconceptions and suggested how we might present our work better. A. G. thanks the University of Catania for the hospitality.The work was partially supported by GNFM of the Italian INDAM (Istituto Nazionale di Alta Matematica).

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A mathematical model of anorexia and bulimia · 2020-03-06 · bulimic; anorexia class, A, in which an individual has the symptoms of anorexia; bulimia class, B, in which an individual