Research ArticleQualitative Stability Analysis of an Obesity Epidemic Modelwith Social Contagion
Enrique Lozano-Ochoa1 Jorge Fernando Camacho1 and Cruz Vargas-De-Leoacuten23
1Maestrıa en Ciencias de la Complejidad Universidad Autonoma de la Ciudad de Mexico 03100 Ciudad de Mexico Mexico2Maestrıa en Ciencias de la Salud Escuela Superior de Medicina Instituto Politecnico Nacional Plan de San Luis SNMiguel Hidalgo Casco de Santo Tomas 11350 Ciudad de Mexico Mexico3Maestrıa en Matematicas Aplicadas Unidad Academica de Matematicas Universidad Autonoma de GuerreroAv Lazaro Cardenas CU 39087 Chilpancingo GRO Mexico
Correspondence should be addressed to Cruz Vargas-De-Leon leoncruz82yahoocommx
Received 1 October 2016 Accepted 13 December 2016 Published 24 January 2017
Academic Editor Tetsuji Tokihiro
Copyright copy 2017 Enrique Lozano-Ochoa et alThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We study an epidemiological mathematical model formulated in terms of an ODE system taking into account both social andnonsocial contagion risks of obesity Analyzing first the case in which themodel presents only the effect due to social contagion andusing qualitative methods of the stability analysis we prove that such system has at the most three equilibrium points one disease-free equilibrium and two endemic equilibria and also that it has no periodic orbits Particularly we found that when considering1198770(the basic reproductive number) as a parameter the system exhibits a backward bifurcation the disease-free equilibrium is stablewhen 1198770 lt 1 and unstable when 1198770 gt 1 whereas the two endemic equilibria appear from 119877lowast
0(a specific positive value reached
by 1198770 and less than unity) one being asymptotically stable and the other unstable but for 1198770 gt 1 values only the former remainsinside the feasible region On the other hand considering social and nonsocial contagion and following the same methodologywe found that the dynamic of the model is simpler than that described above it has a unique endemic equilibrium point that isglobally asymptotically stable
1 Introduction
Obesity has gone from being an isolated health problemrelated to some people to a global problem Considered asldquoThe Pandemia of the 21st Centuryrdquo [1] it is present in bothdeveloped and underdeveloped countries eclipsing in thelatter the problem of malnutrition to become today one ofits main priorities [2] Around the world countries spend ahuge amount of their year budgets as well as qualified humanresources to fight this disease which is frequently associatedwith serious health pathologies such as diabetesmellitus highblood pressure and lung and heart diseases and moreover itis the cause of several kinds of cancer [3 4] Also it affectsthe psychological condition of the individuals because it candamage their self-esteem and social relationships [1 2 5 6]
Clearly obesity is a huge and difficult problem with theaggravating factor that it is present in all sectors of society
regardless of the income ethnicity age gender or anothersocioeconomic status of their individuals Besides it is oftenassociated with the wrong diet sedentary lifestyle or geneticpredisposition of individuals and if it was not enough ithas also been found that obesity can be produced by alarge variety of causes that are linked to cultural social andeconomic conditions of the environment in which peopledevelop Today the way in which the latter influence the ori-gin of obesity is far from being understood In summary theproblem of pandemic obesity is complex and multifactorialand it has increased in recent years over all the world Thusobesity has become a relevant current research topic in whichdifferent fields of human knowledge converge for the purposeof understanding its causes knowing its consequences andas far as possible keeping it under control or eradicating it
In this waymathematicalmodelling is ameans to providea general insight for the dynamics of obesity and as such
HindawiDiscrete Dynamics in Nature and SocietyVolume 2017 Article ID 1084769 12 pageshttpsdoiorg10115520171084769
2 Discrete Dynamics in Nature and Society
could hopefully become a useful device to develop controlstrategies With regard to its causes of social origin thedynamics of obesity can be well modelled by epidemic-typemodels as a process of social contagion as was evidencedby Christakis and Fowler who studied the spread of obesityin a large social network over 32 years and established thatobesity can spread through social ties [7] This approach hasresulted in a wide range of papers of mathematical modellingin which obesity is studied as a social epidemic [8ndash15] Socialobesity epidemic models typically divide the population intotwo or several classes or subpopulations In [12] the classical119878119868119878 model is extended where infection occurs by nonsocialmechanisms as well as through social transmissionThere aremodels in which it has been considered a bilinear incidencerate [14] (for subpopulations of normal weight overweightand obese individuals) obtaining as a result a unique stableequilibrium point in [9ndash11 13] this effect was consideredbut for six subpopulations normal weight latent overweightobese becoming overweight on diet and obese on dietindividuals Other models have incorporated the effects ofthe time delay [15] and have also formulated nonautonomousobesity epidemic models [9] in which periodic positivesolutions were found under some sufficient conditions usinga continuation theorem based on coincidence degree theory
In this paper we analyze the model proposed by Ejima etal [8] a variant of the SIRI model in which the individualswho recover temporarily may get recurrence to infectiousstate and is formulated on the premise that obesity is causedby both social and nonsocial contagion routes [16] Theobjective of this work is to analyze this system using themethods of the qualitative theory of ordinary differentialequations
The rest of this paper is organized as follows in Section 2we present the Ejima et almodel [8] andwe reduce it in a two-dimensional system Section 3 focuses on the case in whichonly the risk of social contagion of obesity is considered weperform a local analysis in order to establish the equilibriumpoints and their corresponding local stabilities as well asbifurcations we also obtain a global analysis by means ofan appropriate Lyapunov function to establish the globalstability of the endemic equilibrium point and by usingDulacrsquos criterion the nonexistence of periodic orbits InSection 4 we will study the case with risk of both socialand nonsocial contagion of obesity we prove the existenceof a unique endemic equilibrium point that is globallyasymptotically stable and also through a suitable Dulacfunction we determine the nonexistence of periodic orbitsSection 5 investigates several important aspects of our modelfrom a numerical point of view Finally in Section 6we collectsome observations and conclusions
2 Obesity Mathematical Model
The model proposed by Ejima et al [8] for the dynamicsof obesity is given by the following set of three differentialequations
119889119878 (119905)119889119905 = 120583119873 minus 120583119878 (119905) minus [120573119868 (119905) + 120598] 119878 (119905)
wherein 119878(119905) 119868(119905) and 119877(119905) respectively denote the suscep-tible (never-obese) infectious (obese) and recovered (ex-obese) individuals in a population In (1) the natural deathand birth rates are assumed to be equal and denoted by 120583thus we have 119878 + 119868 + 119877 = 119873 for all time (the populationsize is constant) Also the parameter 120573 is the transmissionrate due to social contagion risk of obesity 120574 describes therate at which the infectious individuals become recoveredindividuals 120598 is the hazard of obesity due to nonsocialcontagion reasons and 120590 is the relative risk of weight regainamong ex-obese individuals which typically takes a valuegreater than unity (120590 gt 1) due to high risk of coming backto the obese state All the involved parameters are positive
In Figure 1 the flow diagram is shown in which the modelassumptions are based and system (1) is deduced There maybe seen how it is a variant of a SIRI model since the term120590[120573119868(119905) + 120598]119877(119905) given in the second equation (1) representsthe effect of relapse that is the recovered people (ex-obese)after some time become infected (obese) again It should benoted that in total absence of the relapse term (1) is reducedto the corresponding well-known SIRmodel
Adding the three equations (1) we obtain the differentialequation for the total population of this system119889 (119878 + 119868 + 119877)119889119905 = 120583119873 minus 120583 (119878 + 119868 + 119877) (2)
It may be shown easily that the region of biological sense isgiven by
Σ = (119878 119868 119877) isin R3
+ 119878 ge 0 119868 ge 0 119877 ge 0 119878 + 119868 + 119877
= 119873 (3)
that is all solutions starting in Σ remain there for all 119905 ge 0Clearly the set Σ is positively invariant with respect to (1)
21 Reduction to a Two-Dimensional IR System We caneliminate 119878 from the equations of system (1) using the
Discrete Dynamics in Nature and Society 3
identity 119878 + 119868 + 119877 = 119873 in order to obtain the following two-dimensional system written in terms of the variables 119868119877
119889119868 (119905)119889119905 = [120573119868 (119905) + 120598] (119873 minus 119868 (119905) minus 119877 (119905))+ 120590 [120573119868 (119905) + 120598] 119877 (119905) minus (120574 + 120583) 119868 (119905) 119889119877 (119905)119889119905 = 120574119868 (119905) minus 120590 [120573119868 (119905) + 120598] 119877 (119905) minus 120583119877 (119905)
(4)
It is straightforward to prove that the region
Δ = (119868 119877) isin R2
+ 119868 ge 0 119877 ge 0 119868 + 119877 le 119873 (5)
is also a positively invariant set for the reduced model (4)With the aim to carry out a qualitative analysis of the
dynamics for system (4) we study it considering firstly thattransmission only is produced by social contagion route(120598 = 0) and afterwards taking into account both social andnonsocial contagion hazards of the obesity (120598 gt 0) In eachcasewe performed a local and global analysis of the behaviourof solutions of the system
3 Case with Only Social Contagion Risk ofObesity (120598 = 0)
31 Equilibria Local Stability and Backward Bifurcation Ifwe impose the condition 120598 = 0 in system (4) it takes thefollowing form
119889119868 (119905)119889119905 = 120573119868 (119905) (119873 minus 119868 (119905) minus 119877 (119905)) + 120590120573119868 (119905) 119877 (119905)minus (120574 + 120583) 119868 (119905) 119889119877 (119905)119889119905 = 120574119868 (119905) minus 120590120573119868 (119905) 119877 (119905) minus 120583119877 (119905)
(6)
By equalizing to zero the right members of system (6) asis commonly done to find equilibrium points we obtain thecubic polynomial equation
where the coefficients have been defined119886 equiv 1205901205732119887 equiv 120590120573 (120574 + 120583) ( 120583120574 + 120583 + 1120590 minus 1198770) 119888 equiv 120583 (120574 + 120583) (1 minus 1198770)
(8)
In (8) 1198770 is defined as
1198770 equiv 120573119873120574 + 120583 (9)
which is named as the basic reproductive number of thesystem and as is well known is a dimensionless quantitythat represents the average number of secondary infections
BifurcationBifurcationpoint
106 0804 12020
10000
20000
30000
40000
50000
60000
70000
Infe
cted
(I)
Rlowast0
Basic reproductive number (R0)
point R0 = 1
Figure 2 Schematic representation of the bifurcation diagram ofsystem (4) for the case 120598 = 0 Here is shown the variable 119868 as afunction to the parameter 1198770caused by an infective individual introduced into a group ofsusceptible individuals From the second equation (6) 119877lowast isobtained in terms of 119868lowast by means of
According to (7) and (10) we have several points ofequilibrium formodel (6) one disease-free equilibrium1198640 =(0 0) and at least two endemic equilibria 119864lowast = (119868lowast 119877lowast)Besides the solutions of (7) given by 119868lowast = 0 and
wherein 120575 equiv 1198872 minus 4119886119888 is a discriminant can be graphedas functions of 1198770 in the first quadrant of the 1198770119868 planewhere they represent different families of equilibrium pointsone formed by a line of disease-free equilibrium along thehorizontal 1198770 axis and the other two of endemic equilibriumconstituted by quadratic branches In this situation 1198770 playsthe role of a bifurcation parameter See Figure 2
In order to determine the stability of these families ofequilibrium points we need to calculate the Jacobian matrixof system (6) and evaluate it on them This matrix overall isgiven as119869 (119868 119877)
= (120573119873 minus (120574 + 120583) minus 2120573119868 + (120590 minus 1) 120573119877 (120590 minus 1) 120573119868120574 minus 120590120573119877 minus120590120573119868 minus 120583) (12)
Now we perform the local stability analysis consideringseparately the disease-free equilibrium and the endemicequilibria cases
(1) The Disease-Free Equilibrium 1198640 = (0 0) The Jacobianmatrix (12) evaluated in the disease-free equilibrium 1198640 takesthe simple form
whose eigenvalues are 1205821 = minus120583 and 1205822 = minus120583(120574 + 120583)(1 minus 1198770)Note that if 1198770 lt 1 then 1205822 lt 0 and therefore 1198640 is locallyasymptotically stable while when 1198770 gt 1 1205822 gt 0 and 1198640 isunstable In both cases the equilibrium point is hyperbolicThis situation is depicted in Figure 2 by the horizontal line onthe 1198770 axis wherein for 1198770 lt 1we have stability (indicated bya solid line) and for 1198770 gt 1 instability (showed by a dashedline) It should be noted that if 1198770 = 1 then 1205822 = 0 asa consequence the equilibrium point is nonhyperbolic and1198770 = 1 could be the value in which a bifurcation is producedHence we have the following result
Theorem 1 If 1198770 lt 1 then 1198640 is an equilibrium of system (6)and it is locally asymptotically stable Otherwise if1198770 gt 1 then1198640 is unstable(2)The Endemic Equilibria 119864lowast = (119877lowast 119868lowast) By substituting theendemic equilibria 119864lowast = (119868lowast 119877lowast) in system (6) the identitiesare obtained (120590 minus 1) 120573119877lowast = 120573119868lowast minus 120573119873 + (120574 + 120583)
120583119877lowast = 120574119868lowast minus 120590120573119868lowast119877lowast (14)
which allows writing the Jacobian matrix (12) evaluated insuch points as
119869 (119868lowast 119877lowast) = (minus120573119868lowast (120573119868lowast minus (120574 + 120583) (1 minus 1198770)) 119868lowast119877lowast120583119877lowast119868lowast minus120590120573119868lowast minus 120583 ) (15)
Det (119869) equiv 119886119868lowast2 minus 119888 (18)
are the trace and determinant of (15) respectively Becausein the Δ region both coordinates of 119864lowast are positive thenTr(119869) always is negative not so with Det(119869) which could bepositive or negative According to the discriminant 120575 of (11)in the corresponding analysis of the equilibrium points inorder to ensure that 119864lowast isin Δ the following three cases maybe identified
(i) If 120575 lt 0 then considering in this condition expres-sions 119886 119887 and 119888 given by (8) we obtain 1198770 lt 119877lowast
0
where
119877lowast0equiv (120590 minus 1) 120583 + 120574 + 2radic(120590 minus 1) 120583120574120590 (120574 + 120583) (19)
Consequently in accordance with (11) in this intervalwe have that there is not endemic equilibrium pointsSince 120583 and (120590minus1)120574 are two positive real numbers the
relation between its arithmetic and geometric meansis always given by
12 [(120590 minus 1) 120574 + 120583] ge radic(120590 minus 1) 120574120583 (20)
therefore from (19) and (20) itmay be concluded that119877lowast0le 1
(ii) If 120575 = 0 then substituting 119886 119887 and 119888 given by (8)in this equality we obtain 1198770 = 119877lowast
0 Thus according
to (11) when 1198770 reaches this value we have only oneendemic equilibrium point and 119868lowast = 119868lowast
0 where
119868lowast0equiv radic120574120583 (120590 minus 1) minus 120583120590120573 (21)
is the corresponding infected population of119877lowast0More-
over if (19) and (21) are substituted in determinant(18) and trace (17) these are simplified as
Det (119869) = 0 (22)
Tr (119869) = minus(120590 + 1)radic120574120583 (120590 minus 1) minus 120583120590 (23)
respectively Consequently taking into account (22)and (23) eigenvalues (16) are reduced to 1205821 = Tr(119869) lt0 and 1205822 = 0 one of which is negative and theother is zero Therefore this equilibrium point isnonhyperbolic and 1198770 = 119877lowast
0could also be the value
in which a bifurcation is produced See Figure 2(iii) If 120575 gt 0 then similarly using 119886 119887 and 119888 given by (8)
this inequality leads to 119877lowast0lt 1198770 As a consequence
from (11) it follows that in this interval we havefamilies of endemic equilibrium points determinedby two quadratic branches It may be shown that forall endemic equilibrium points that form the uppercurve since 119868lowast gt 119868lowast
0 from (18) it results in the fact
that Det(119869) gt 0 thereby in (16) it follows that botheigenvalues 1205821 and 1205822 are negative and thereforethese equilibriumpoints are asymptotically stable Onthe other hand it can also be shown that for endemicequilibrium points that are part of the lower curvesince 119868lowast lt 119868lowast
0 from (18) it occurs that Det(119869) lt 0
in this way again from (16) it is obtained that 1205821 gt 0and1205822 lt 0 and consequently such equilibriumpointsare unstable Note that in the latter case (in orderthat 119864lowast isin Δ) quadratic branch is delimited by thepoint (1 0) in which by the way Det(119869) = 0 andtherefore 1205821 lt 0 and 1205822 = 0 that is the said point isnonhyperbolic and in it a bifurcation could also occurThese results are illustrated in Figure 2
It is necessary to point out that from the above arguments itfollows that if 0 lt 1198770 lt 119877lowast
0 then the disease-free family of
equilibrium points 1198640 is the only one in the feasible regionΔ Based on the previous analysis we have the followingtheorem which summarizes the local stability of all endemicequilibria points of system (6) contained in Δ
Discrete Dynamics in Nature and Society 5
Theorem 2 If 0 lt 1198770 lt 119877lowast0 then there are not endemic
equilibrium points 119864lowast in the feasible region Δ Also If 1198770 =119877lowast0 then in such region there is a unique 119864lowast which is not
hyperbolic Moreover if 119877lowast0
lt 1198770 then there are familiesof endemic equilibrium points determined by two quadraticbranches an upper in which all points are asymptoticallystable and the other lower formed by unstable points and withepidemiological significance only when 1198770 ⩽ 1
In Figure 2 the results indicated by Theorems 1 and 2are displayed schematically There it may be appreciated thatthe trajectories of solutions of system (4) change abruptly inthe nonhyperbolic points (119877lowast
0 119868lowast) and (1 0) that is in these
points really bifurcations occur In that sense Figure 2 can beconsidered as the bifurcation diagram in the feasible regionΔ of said system and the type of bifurcation shown in itcorresponds to a backward bifurcation
32 Nonexistence of Periodic Orbits and Global StabilityGlobal analysis of system (6) includes the study of periodicorbits The following result shows that this system does nothave periodic orbits
Theorem 3 System (6) does not have periodic orbits in theinterior of ΔProof Consider system (6) for 119868 gt 0 and 119877 gt 0 We considerthe Dulac function
that is 120597(Φ119875)120597119868 + 120597(Φ119876)120597119877 lt 0 in the interior of Δit follows from the Dulac criterion that system (6) has noperiodic orbits in that region
On the other hand global analysis of system (6) alsoinvolves the study of global stability of its disease-freeequilibrium and its endemic equilibria The former may beproved by arguments of local stability as can be seen in thefollowing theorem
Theorem 4 If 0 lt 1198770 lt 119877lowast0 then the unique disease-free
equilibrium 1198640 of system (6) is globally asymptotically stablein ΔProof According to the discussion given previously duringthe formulation of Theorems 1 and 2 it was found that if0 lt 1198770 lt 119877lowast
0 then the disease-free equilibrium point 1198640
is the only one and it also is asymptotically stable in thefeasible region Δ Besides according to Theorem 3 there arenot periodic orbits here As a result all initial condition inΔ which satisfies this interval originates paths tending to1198640 consequently such point must be globally asymptoticallystable
Finally in order to prove the global stability of a uniqueendemic equilibrium 119864lowast in the interior of Δ when 1198770 gt 1 weuse the method of Lyapunov functions This is accomplishedin the next theorem
Theorem5 Assume that1198770 gt 1Then a unique endemic equi-librium 119864lowast = (119868lowast 119877lowast) of system (6) is globally asymptoticallystable in the interior of ΔProof To prove the global asymptotic stability of the uniqueendemic equilibrium 119864lowast we define a Lyapunov function119882(119868 119877) isin R2
0+ 119868 gt 0 119877 gt 0 rarr R given by
119882(119868 119877) = 120583(ln 119868119868lowast + 119868lowast119868 minus 1)+ 120573 (120590 minus 1) (119877 minus 119877lowast minus 119877lowast ln 119877119877lowast )
(26)
Let119882119894 = ln(119868119868lowast) + 119868lowast119868 minus 1 By using 120573119873 minus (120574 + 120583) = 120573119868lowast minus120573(120590 minus 1)119877lowast we have119889119882119894119889119905 = 120573119868lowast (2 minus 119868lowast119868 minus 119868119868lowast)
+ 120573 (120590 minus 1) 119877lowast ( 119877119877lowast minus 1 minus 119868lowast119877119868119877lowast + 119868lowast119868 ) (27)
Let119882119903 = 119877minus119877lowastminus119877lowast ln(119877119877lowast) By using 120574 = 120590120573119877lowast+120583(119877lowast119868lowast)we have
119889119882119903119889119905 = 120590120573119877lowast119868 (2 minus 119877119877lowast minus 119877lowast119877 )+ 120583119877lowast ( 119868119868lowast minus 119877119877lowast minus 119868119877lowast119868lowast119877 + 1) (28)
The derivative of (26) along solution of (6) is given by
119889119882119889119905 = 120583119889119882119894119889119905 + 120573 (120590 minus 1) 119889119882119903119889119905 (29)
By means of (27) and (28) we obtain
119889119882119889119905 = 120583120573 (119868lowast minus (120590 minus 1) 119877lowast) (2 minus 119868lowast119868 minus 119868119868lowast)+ 120573 (120590 minus 1) 120590120573119877lowast119868 (2 minus 119877119877lowast minus 119877lowast119877 )+ 120583120573 (120590 minus 1) 119877lowast (2 minus 119868119877lowast119868lowast119877 minus 119868lowast119877119868119877lowast ) lt 0
(30)
Using
120573119868lowast minus 120573 (120590 minus 1) 119877lowast = 120573119873 minus (120574 + 120583)= (120574 + 120583) (1198770 minus 1) (31)
6 Discrete Dynamics in Nature and Society
we have119889119882119889119905 = 120583 (120574 + 120583) (1198770 minus 1) (2 minus 119868lowast119868 minus 119868119868lowast)+ 120573 (120590 minus 1) 120590120573119877lowast119868 (2 minus 119877119877lowast minus 119877lowast119877 )+ 120583120573 (120590 minus 1) 119877lowast (2 minus 119868119877lowast119868lowast119877 minus 119868lowast119877119868119877lowast ) lt 0
(32)
Therefore if 1198770 gt 1 then 119889119882119889119905 is negative definite Bythe Lyapunov asymptotic stability theorem [17] this resultimplies that119864lowast is globally asymptotically stable in the interiorof ΔRemark 6 Recently Vargas-De-Leon [18] used the Lyapunovfunction119882119894 = ln(119868119868lowast)+119868lowast119868minus1 to prove global stability of thecoexistence equilibrium of two-species mutualism models
4 Case with Social and Nonsocial ContagionRisks of Obesity (120598 gt 0)
41 Equilibria and Nonexistence of Periodic Orbits Wemightfollow the usual method of setting the right-hand side of (4)equal to zero in order to obtain their equilibria This proce-dure is not the most appropriate because it leads to a cubicequation whose analytical solutions are quite complicatedInstead an alternative way to find the equilibrium points isto perform a geometric analysis based on the intersection ofthe nullclines of system (4) The nullclines = 0 (verticaldirections) and = 0 (horizontal directions) are given bythe functions
1198661 (119868) = 1205731198682 minus (120573119873 minus 120598 minus 120583 minus 120574) 119868 minus 120598119873120573 (120590 minus 1) 119868 + 120598 (120590 minus 1) (33)
respectivelyNote that1198661(119868) and1198662(119868) respectively are discontinuous
in 119868 = minus120598120573 and 119868 = minus(120590120598 + 120583)120590120573 Besides it is clear that1198661(119868) rarr +infin (resp 1198662(119868) rarr 120574120590120573) as 119868 rarr +infin and that thefunction1198661(119868) (resp1198662(119868)) is increasing and concave downGiven that 1198661(0) = minus119873(120590 minus 1) lt 0 and 1198662(0) = 0 thenthe functions 1198661(119868) and 1198662(119868) intersect at a single point (theequilibrium point) in the first quadrant (see Figure 3) Thisresult indicates that in Δ an endemic equilibrium point existsand is unique
Regarding whether or not system (4) has periodic orbitswe have precisely the following result
Theorem 7 System (4) does not have periodic orbits in theinterior of ΔProof Consider system (4) for 119868 gt 0 and 119877 gt 0 From theDulac function
Φ (119868 119877) = 1119868 (35)
Equilibrium point
minus120590120576 + 120583
120590120573minus120576
120573
minusN
120590 minus 1
120574
120590120573
R
G1(I)
G2(I)
I
Figure 3 Schematic representation of a branch of the verticalnullcline 1198661(119868) (dashed line) and the horizontal nullcline 1198662(119868)(solid line) Both curves intersect inΔ once at the equilibrium point
Equilibriumpoint
R
(0 N)
R = N minus I
I+ (N 0) I
Figure 4 Trapping region bounded by the dashed line formed bythe horizontal and vertical axes and the diagonal line 119877 = 119873 minus 119868This triangular region contains inside the only equilibrium point
it follows that120597 (Φ119875 (119868 119877))120597119868 + 120597 (Φ119876 (119868 119877))120597119877= minus 1205981198682 (119873 minus (119868 + 119877)) minus (120573 + 120598119868) minus 1205901205981198771198682minus 120590(120573 + 120598119868) minus 120583119868
(36)
Since 120597(Φ119875)120597119868+120597(Φ119876)120597119877 lt 0 inΔ taking into account theDulac criterion we conclude that system (4) has no periodicorbits
42 Local andGlobal Stability In order to determine the localand global stability of the single equilibrium point in Δ weconsider that it is inside of the triangular region bounded bythe dashed line shown in Figure 4 This is a trapping region
Discrete Dynamics in Nature and Society 7
the vector field of system (4) on the boundary point into thebox For the points on the horizontal side = 120574119868 gt 0 and = minus(119868 minus 119868+)(119868 minus 119868minus) where 119868+ and 119868minus are the positive andnegative roots of (120573119868 + 120598)(119868 minus 119873) + (120574 + 120583)119868 respectivelyThus these inequalities imply that the vector field points upand right on the interval (0 119868+) while it goes up and left on(119868+ 119873) Moreover in the vertical side = minus(120590120598 + 120583)119877 le 0and = 120598119873 + 120598(120590 minus 1)119877 gt 0 This implies that the vector fieldis directed downward and right except at the origin where itgoes to the right
Finally on the diagonal side of slope minus1 extending fromthe point (0119873) to (119873 0) it can be shown that
119889119877119889119868 = minus1 + 120583119873(119868 minus 119868lowast+) (119868 minus 119868lowast
minus) (37)
where 119868lowast+and 119868lowastminusare the positive and negative roots of 120590(120573119868 +120598)(119868 minus 119873) + (120574 + 120583)119868 respectively Hence the vector field
for large values of 119868 is almost parallel to the diagonal lineIn a more precise analysis always 119868 minus 119868lowast
minusgt 0 and 119868 minus 119868lowast
+gt0 if 119868 gt 119868lowast
+or 119868 minus 119868lowast
+lt 0 if 119868 lt 119868lowast
+ According to
(37) the first case implies that the vector field points inward(to the right) on the diagonal line because 119889119877119889119868 is morenegative than minus1 while the latter indicates that the vectorfield is directed inside (to the left) the diagonal side because119889119877119889119868 is less negative than minus1 (See Figure 4) Therefore weconclude that the triangular region is effectively a trappingregion
On the other hand Theorem 7 prohibits the existenceof closed orbits inside the triangular region Consequentlythe trajectories entering the triangular region accordingto the Poincare-Bendixson theorem [19] must converge tothe single equilibrium point located inside it Thereforethe equilibrium point inside the triangular region must beasymptotically stable Moreover since this point is the onlypoint of equilibrium in Δ it is also globally asymptoticallystable
The results of this section may be summarized in thefollowing theorem
Theorem 8 System (4) has a unique equilibrium point in thetriangular region Δ which is globally asymptotically stable
5 Numerical Results
In this section we perform a series of numerical simulationsand graphs of system (4) to illustrate the different resultsobtained for each of the two cases of interest 120598 gt 0 and120598 = 0 previously analyzed We will use the parameter valuesreported in the work developed by Ejima et al [8] theirepidemiological meanings and magnitudes are indicated inTable 1
51 Case 120598 = 0 According to the magnitudes of theparameters reported in Table 1 (19) takes the value 119877lowast
0=069363 whereas (21) acquires the value 119868lowast
0= 163304
Table 1 Baseline values and epidemiological meanings of theparameters given by Ejima et al [8]
Parameter Baseline values Epidemiological meaning
119873 100000 individuals(assumed) Population size
1120583 694 (per year) Average life expectancy at birth120573 296 times 10minus7 (peryear)
Transmission rate of obesity dueto social contagion120598 0012 (per year) Nonsocial contagion risk ofobesity120590 80 Relative hazard of obesity amongthe ex-obese1120574 358 (per year) Average duration of obesity
2000
20000
0
4000
4000
6000
600080
00
8000
1000
0
10000
1200
0
12000
1400
0
14000
1600
0
16000
1800
0
18000
2000
0
200002200024000260002800030000
2200
024
000
2600
028
000
3000
0
I (obese)
R (e
x-ob
ese)
Figure 5 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 03 obtained by considering 120573 = 127 times 10minus7Furthermore taking into consideration (10) and (11) theresulting two endemic equilibria are119864lowast
1= (20918 91378) (38)
119864lowast2= (12522 79384) (39)
which as discussed previously are asymptotically stable andunstable respectively
In Figures 5ndash9 we illustrated representative phase por-traits of the dynamic system (6) for some different values of1198770 between 0 and just over 1 according to the bifurcationdiagram shown in Figure 2 These values are obtained bymaintaining fixed in (9) 119873 120583 and 120574 but varying theparameter 120573 Thus if 1198770 takes values between 0 and 119877lowast
0
say 1198770 = 03 we have only one equilibrium point (thedisease-free equilibrium) which is globally asymptoticallystable (see Figure 5) If 1198770 = 119877lowast
0 we have two equilibrium
points one asymptotically stable disease-free equilibriumand the other endemic equilibria point which is nonhyper-bolic (see Figure 6) When 1198770 takes values between 119877lowast0 and
8 Discrete Dynamics in Nature and Society
2000
2000
0
0
4000
4000
6000
6000
8000
800010
000
10000
1200
0
12000
1400
0
14000
1600
0
16000
1800
0
18000
2000
0
20000
2200
0
2400
0I (obese)
R (e
x-ob
ese)
Figure 6 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 119877lowast0 obtained by considering 120573 = 294 times 10minus7
5000
2000
0
0
1000
0
4000
1500
0
6000
2000
0
8000
2500
0
10000
3000
0
12000
3500
0
14000
4000
0
16000
4500
0
18000
5000
0
20000
I (obese)
R (e
x-ob
ese)
Figure 7 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 08 obtained by considering 120573 = 339 times 10minus71 say 1198770 = 08 we have three equilibria one disease-free equilibrium (asymptotically stable) and two endemicequilibria (asymptotically stable and unstable) as shown inFigure 7 If 1198770 = 1 there are two equilibrium pointsone disease-free equilibrium which is nonhyperbolic and theother asymptotically stable endemic equilibrium as shown inFigure 8 Finally if 1198770 takes a slightly greater value to 1 forexample1198770 = 12 we have again two equilibrium points onedisease-free equilibrium (unstable) and the other endemicequilibrium (asymptotically stable) but both are hyperbolic(see Figure 9)
On the other hand in Figures 10ndash14 examples of theevolution in time of 119868(119905) and119877(119905) are presented for the values1198770 = 03 1198770 = 119877lowast
0 1198770 = 08 1198770 = 1 and 1198770 = 12 which
1000
00
2000
0
3000
0
4000
0
5000
0
6000
0
7000
0
8000
0
9000
0
1000
00
1100
00
1200
00
I (obese)
10000
20003000400050006000700080009000
10000110001200013000140001500016000
R (e
x-ob
ese)
Figure 8 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 1 obtained by considering 120573 = 423 times 10minus7
1000
0
10000
0
2000
0
2000
3000
0
30004000
4000
050
000
50006000
6000
07000
7000
0
80009000
8000
090
000
1000011000
1000
00
12000
1100
00
1400013000
1200
0013
0000
1400
00
1500016000
I (obese)
R (e
x-ob
ese)
Figure 9 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 12 obtained by considering 120573 = 508 times 10minus7illustrate respectively some of the dynamics shown in thephase portraits in Figures 5ndash9
52 Case 120598 gt 0 Taking into account the value 120598 =0012 gt 0 given in Table 1 in addition to the othersthere reported nullclines (33) and (34) intersect at the onlyendemic equilibrium point119864lowast = (60890 66805) (40)
which is globally asymptotically stable In Figure 15 thecorresponding phase portrait with several representativetrajectories around this equilibrium point is shown Also
Discrete Dynamics in Nature and Society 9
Time (years)
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 10 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 03 and the initial condition is(119868(0) 119877(0)) = (10000 30000)
Time (years)0 100 200 300
ObeseEx-obese
20000
15000
10000
5000
0
Figure 11 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 119877lowast
0and the initial condition is(119868(0) 119877(0)) = (20000 8000)
performing another series of numerical simulations of system(4) we show for this case in Figure 16 an example of theevolution in time of 119868(119905) and 119877(119905)6 Discussion and Conclusions
According to the results of our analysis of the Ejima et almodel we arrive at the following conclusions It has beenshown that the Ejima et al model [8] if only the socialcontagion (120598 = 0) is considered presents the two bifurcationvalues 1198770 = 119877lowast
0and 1198770 = 1 which results in what is
known as backward bifurcation From a mathematical pointof view the backward bifurcation is apparently constitutedby the combination of two different kinds of bifurcations
Time (years)
40000
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 12 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 08 and the initial condition is(119868(0) 119877(0)) = (30000 12000)
Time (years)
40000
50000
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 13 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 1 and the initial condition is(119868(0) 119877(0)) = (18000 2500)a saddle-node bifurcation that occurs at 1198770 = 119877lowast
0 whose
upper (asymptotically stable) and lower (unstable) quadraticbranches are defined for 119877lowast
0lt 1198770 and a transcritical
bifurcation that takes place in 1198770 = 1 whose horizontalbranches on the 1198770 axis are asymptotically stable for 1198770 lt 1and unstable for 1 lt 1198770 For values of 1 lt 1198770 only theupper quadratic branch asymptotically stable of the saddle-node bifurcation and the unstable horizontal branch of thetranscritical bifurcation have ameaning in the feasible regionΔ The diagram shown in Figure 2 suggests the presenceof these two bifurcations in the vicinities of the points(119877lowast0 119868lowast0) and (1 0) Of course we should go beyond the
graphic evidence and employ the qualitative analysis of localbifurcations in those points to verify our statement even
10 Discrete Dynamics in Nature and Society
Time (years)0 100 200 300
ObeseEx-obese
140000
120000
100000
80000
60000
40000
20000
0
Figure 14 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 12 and the initial condition is(119868(0) 119877(0)) = (100000 80000)
1000
0
10000
0
2000
0
2000
3000
0
30004000
4000
0
5000
0
50006000
6000
0
7000
7000
0
80009000
8000
0
9000
0
1000011000
1000
00
12000
1100
00
1400013000
1200
00
1300
00
1400
00
1500016000
I (obese)
R (e
x-ob
ese)
Figure 15 Phase portrait for system (4) for the case 120598 = 0012 gt 0(considering 120573 = 296 times 10minus7)without the realization of this depth analysis our discussionsand conclusions can be supported firmly from the resultsshown in this work of investigation
Indeed the presence of the different branches of familiesof equilibrium points shown in Figure 2 causes the system todisplay as the parameter1198770 is varied a bistable region and asa consequence the hysteresis phenomenon In order to showin more detail the presence of hysteresis in the bifurcationdiagram mentioned before suppose we start the system inthe state119860 (see Figure 17) and then slowly increase 1198770 (whichis indicated by an arrow to the right just beneath the axis1198770 of Figure 17) We remain at the origin until 1198770 = 1when the origin loses its stability At this point at the slightestdisturbance the system will jump to the state 119861 located in the
0
20000
40000
60000
80000
100000
120000
Time (years)0 100 200 300
ObeseEx-obese
Figure 16 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0012 gt 0 and the initial condition is(100000 50000)
B
A
1
C
80000
70000
60000
50000
40000
30000
20000
10000
0
Infe
cted
(I)
Rlowast0
Basic reproductive number (R0)
Figure 17 Schematic representation of hysteresis phenomenon thatexhibits the bifurcation diagram of system (4) for the case 120598 = 0Here the forward and backward paths are not equal as the parameter1198770 is changedstable upper branch Insofar as1198770 increases its value (1198770 gt 1)the equilibrium points of the system will move along thisbranch away from the state119861 If now1198770 continuously reducedits value equilibrium states will return to the state 119861 cross itand reach the state 119862 (in which 1198770 = 119877lowast
0) At this point at
the slightest change again equilibrium states will jump backto the origin (in 1198770 = 119877lowast0 ) and if 1198770 continues to decrease wewill return to the state119860 Consequently the system exhibits alack of reversibility or it is said that it has memory (or that itpresents hysteresis) because the forward and backward pathsare not identical
From an epidemiological point of view in this backwardbifurcation if 0 lt 1198770 lt 119877lowast
0 no matter where the initial
Discrete Dynamics in Nature and Society 11
conditions are taken solutions always tend to the straightline of stable disease-free points (that correspond to a never-obese population) When 119877lowast
0lt 1198770 lt 1 if the initial
conditions are located below the unstable quadratic branchof endemic equilibrium points (obese people) then solutionscontinue going to the horizontal line of stable disease-freepoints otherwise they will be directed to the stable curveof endemic points located at the upper quadratic branchFinally if 1 lt 1198770 no matter again where the initialconditions are situated solutions will continue heading forthe stable endemic points of such upper branch Should benoted that unlike what happens in systems that exhibit thephenomenon of forward bifurcation in which the endemicequilibrium exists only for 1198770 gt 1 [20] in systems exhibitinga backward bifurcation under certain initial conditions itis possible to have endemic states when 1198770 lt 1 In ourcase the endemic equilibrium that exists for 1198770 just aboveone has a large obese population so the result of 1198770 risingabove one would be a sudden and dramatic jump in thenumber of infective Moreover reducing 1198770 back below onewould not eradicate the obesity if the size of the infectedpopulation is greater or equal than 119868lowast
0 the corresponding
value of 119877lowast0 In order to eradicate the disease 1198770 must be
further reduced so it just needs to be slightly less than119877lowast0 to suddenly enter the region where only stable disease-
free equilibrium (never-obese population) exists Both suchdifferent abrupt behaviours are the result of hysteresisphenomenon
It can be seen from the bifurcation diagram shown inFigure 2 that theway inwhich obesity infected population canbe lowered (or raised) is through the reduction (or increase)of 1198770 parameter which in accordance with (9) depends onthe four quantities 119873 120573 120583 and 120574 Since we have assumedin (1) that the natural death and birth rates are equal anddenoted by 120583 the population size119873 is constantTherefore1198770can only change according to the following forms Firstly 1198770diminishes if the natural rate 120583 (of death or birth) increasesthe transmission rate 120573 decreases or the rate 120574 grows atwhich the infected individuals become recovered On theother hand 1198770 increment its value if the opposite occurs 120583decreases 120573 goes up or 120574 declines It should be noted thatchanges in 120583 could be slower and achieved in the long termprobably in decades which is not a viable option Converselychanges in 120573 or 120574 might be faster and perhaps occur infewer years Since the focus is on reducing obesity levels andtherefore the values of 1198770 it is of interest to seek mechanismsto decrease120573 andor increase 120574The former could be achievedby sensitizing at the population in general about the benefitsassociatedwith having a healthy and balanced diet and also bydesigning effective health and informative public preventionpoliciesThe latter could be done by providing better medicalassistance and making it available to the obese populationIt should be emphasized that this last strategy is not feasiblein countries with high levels of obesity as Mexico because atthe present time it is very expensive and will become evenmore expensive as the affected population increases In thisregard we believe that it is much more convenient that thegovernments of the affected countries implement broaderand more effective preventive policies
It is worth mentioning that in Mexico there have beenstudies to quantify more precisely the prevalence of peoplesuffering from overweight and obesity It is estimated that713 of Mexican adults aged 20 or more are found inthis condition [21] While trends show slowing down of theincrease in the obesity prevalence there is no evidence toinfer that prevalence will decrease in the coming years Inthis sense studies like this allow us to analyze the trends ofthe population with obesity and study the different possiblescenarios Particularly with the model studied this couldbe done for Mexico if we knew accurately the values ofits parameters 120573 and 120574 To our knowledge there is notsuch information Nevertheless these parameters could beestimated in a short or medium term based on statisti-cal information collected on obesity in our country Theknowledge of how these parameters behave over time wouldcontribute in quantifying the effectiveness of the Mexicanpublic policies against obesity
With respect to the model studied it is pertinent tocomment about the term of relapse 120590[120573119868(119905) + 120598]119877(119905) of thesecond equation (1) In the first place in this work it has beenconsidered that the parameter 120590 the relative risk of weightregain among ex-obese individuals is greater than unity (120590 gt1)This condition represents the high risk that recovered (ex-obese) individuals will become obese again However it isalso plausible to consider that 0 lt 120590 lt 1 which representsthat ex-obese population is more resistant to being obeseagain that is they are more aware of the health risks thatthis would represent The consideration that 0 lt 120590 lt 1 doesnot change the results of the stability analysis and periodicorbits performed in Sections 3 and 4 With respect to thelatter the nullcline 1198661 (given by (33)) changes to 1198661015840
1 where1198661015840
1= minus1198661 while 1198662 (34) remains the same The intersection
of the nullclines 11986610158401and 1198662 again occurs at only one point
in the region Δ and its stability is the same Regarding thisaspect we could conclude that regardless of the value ofthe parameter 120590 there is always the possibility that the ex-obese population is at risk of relapse and becoming obeseagain of course this possibility is less if 0 lt 120590 lt 1 andgreater if 120590 gt 1 In this sense public health programs againstobesity designed by governments may be more effectiveif they take into account the awareness of the ex-obesepopulation
Finally with regard to the mentioned relapse term120590[120573119868(119905) + 120598]119877(119905) if we consider only the effects of socialcontagion (which corresponds to 120598 = 0) it is reduced tothe nonlinear term 120590120573119868(119905)119877(119905) which causes system (4) toexhibit a backward bifurcation If the effects of nonsocialcontagion (that is 120598 gt 0) are also incorporated the presenceof the linear term 120590120573120598119877(119905) causes the backward bifurcationto disappear in system (4) and in its place it only presentsan equilibrium point global asymptotically stable The lattercase suggests that the way in which the effects of nonsocialcontagion are introduced into the model by means of thelinear part of the relapse term should be modified in orderto maintain much of the interesting dynamic that is presentin the backward bifurcation Certainly an alternative wayis considering instead of a linear a nonlinear contribution
12 Discrete Dynamics in Nature and Society
Perhaps it is possible to build amore general nonlinearmodelwhere in the case in which the contribution due to nonsocialeffects gradually fades the backward bifurcation of the Ejimaet al model could be obtained as a limit case
Competing Interests
The authors declare that they have no conflict of interests
Acknowledgments
This research was supported by the UACM-SECITI ResearchGrant PI-2013-34 Cruz Vargas-De-Leon would like toexpress his indebtedness to Dr Ramon Reyes Carreto for hishospitality during his stay as visiting professor at the UAGroin Chilpancingo
References
[1] World Health Organization Obesity Preventing and Managingthe Global Epidemic Report of a WHO Consultation WHOTechnical Report Series No 894 WHO Geneva Switzerland2000 httpswwwgooglecommxurlsa=tamprct=jampq=ampesrc=sampsource=webampcd=1ampved=0ahUKEwjlno 3tsDRAhVqi1QKH-W0rBPIQFggZMAAampurl=http3A2F2Fwhqlibdocwhoint2Ftrs2FWHO TRS 894pdfampusg=AFQjCNFLusNXrJz-xFKX1xpQwkHLEr-Lohgampbvm=bv144224172damcampcad=rja
[2] C L Vanvrancken-Tompkins andMaM Sothern ldquoPrevencionde la obesidad en los ninos desde el nacimiento hasta loscincoanosrdquo in Enciclopedia Sobre el Desarrollo de la PrimeraInfancia R Tremblay R Barr R Peters and M Boivin Edsvol 1ndash7 Centre of Excellence for EarlyChildhoodDevelopmentQuebec Canada 2010
[3] S K Garg H Maurer K Reed and R Selagamsetty ldquoDiabetesand cancer two diseases with obesity as a common risk factorrdquoDiabetes Obesity andMetabolism vol 16 no 2 pp 97ndash110 2014
[4] National Heart ldquoLung and Blood Institute World Health WhatAre the Health Risks of Overweight and Obesityrdquo httpswwwnhlbinihgovhealthhealth-topicstopicsoberisks
[5] World Health Organization (WHO) Global Health Observa-tory Data Repository WHO Geneva Switzerland 2011 httpappswhointghodata
[6] World Health Organization The Global Strategy on Diet Phys-ical Activity and Health (DPAS) WHO Geneva Switzerland2004 httpwwwwhointnmhwha59dpasen
[7] N A Christakis and J H Fowler ldquoThe spread of obesity in alarge social network over 32 yearsrdquoThe New England Journal ofMedicine vol 357 no 4 pp 370ndash379 2007
[8] K Ejima K Aihara and H Nishiura ldquoModeling the obesityepidemic social contagion and its implications for controlrdquoTheoretical Biology and Medical Modelling vol 10 article 17 pp1ndash13 2013
[9] A J Arenas G Gonzalez-Parra and L Jodar ldquoPeriodic solu-tions of nonautonomous differential systems modeling obesitypopulationrdquo Chaos Solitons amp Fractals vol 42 no 2 pp 1234ndash1244 2009
[10] G Gonzalez-Parra L Acedo R-J Villanueva Mico and A JArenas ldquoModeling the social obesity epidemic with stochasticnetworksrdquo Physica A Statistical Mechanics and its Applicationsvol 389 no 17 pp 3692ndash3701 2010
[11] G Gonzalez-Parra L Jodar F J Santonja and R J VillanuevaldquoAn age-structured model for childhood obesity mathematicalpopulation studiesrdquo An International Journal of MathematicalDemography vol 17 no 1 pp 1ndash11 2010
[12] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 Article ID e1000968 2010
[13] L Jodar F J Santonja and G Gonzalez-Parra ldquoModelingdynamics of infant obesity in the region of Valencia SpainrdquoComputers andMathematics with Applications vol 56 no 3 pp679ndash689 2008
[14] F-J Santonja R-J Villanueva L Jodar and G Gonzalez-Parra ldquoMathematical modelling of social obesity epidemic inthe region of Valencia Spainrdquo Mathematical and ComputerModelling of Dynamical Systems vol 16 no 1 pp 23ndash34 2010
[15] F J Santonja and L Shaikhet ldquoProbabilistic stability analysisof social obesity epidemic by a delayed stochastic modelrdquoNonlinear Analysis Real World Applications vol 17 pp 114ndash1252014
[16] D Tudor ldquoA deterministic model for herpes infections inhuman and animal populationsrdquo SIAM Review vol 32 no 1pp 136ndash139 1990
[17] A M Lyapunov The general problem of the stability of motionTaylor amp Francis London England 1992
[18] C Vargas-De-Leon ldquoLyapunov functions for two-species coop-erative systemsrdquo Applied Mathematics and Computation vol219 no 5 pp 2493ndash2497 2012
[19] D W Jordan and P Smith Nonlinear Ordinary DifferentialEquations An Introduction for Scientists and Engineers OxfordUniversity Press New York NY USA 4th edition 2007
[20] C M Kribs-Zaleta and J X Velasco-Hernandez ldquoA simplevaccination model with multiple endemic statesrdquoMathematicalBiosciences vol 164 no 2 pp 183ndash201 2000
[21] S Barquera I Campos-Nonato L Hernandez-Barrera APedroza-Tobıas and J A Rivera-Dommarco ldquoPrevalence ofobesity in mexican adults ensanut 2012rdquo Salud Publica deMexico vol 55 supplement 2 pp S151ndashS160 2013
could hopefully become a useful device to develop controlstrategies With regard to its causes of social origin thedynamics of obesity can be well modelled by epidemic-typemodels as a process of social contagion as was evidencedby Christakis and Fowler who studied the spread of obesityin a large social network over 32 years and established thatobesity can spread through social ties [7] This approach hasresulted in a wide range of papers of mathematical modellingin which obesity is studied as a social epidemic [8ndash15] Socialobesity epidemic models typically divide the population intotwo or several classes or subpopulations In [12] the classical119878119868119878 model is extended where infection occurs by nonsocialmechanisms as well as through social transmissionThere aremodels in which it has been considered a bilinear incidencerate [14] (for subpopulations of normal weight overweightand obese individuals) obtaining as a result a unique stableequilibrium point in [9ndash11 13] this effect was consideredbut for six subpopulations normal weight latent overweightobese becoming overweight on diet and obese on dietindividuals Other models have incorporated the effects ofthe time delay [15] and have also formulated nonautonomousobesity epidemic models [9] in which periodic positivesolutions were found under some sufficient conditions usinga continuation theorem based on coincidence degree theory
In this paper we analyze the model proposed by Ejima etal [8] a variant of the SIRI model in which the individualswho recover temporarily may get recurrence to infectiousstate and is formulated on the premise that obesity is causedby both social and nonsocial contagion routes [16] Theobjective of this work is to analyze this system using themethods of the qualitative theory of ordinary differentialequations
The rest of this paper is organized as follows in Section 2we present the Ejima et almodel [8] andwe reduce it in a two-dimensional system Section 3 focuses on the case in whichonly the risk of social contagion of obesity is considered weperform a local analysis in order to establish the equilibriumpoints and their corresponding local stabilities as well asbifurcations we also obtain a global analysis by means ofan appropriate Lyapunov function to establish the globalstability of the endemic equilibrium point and by usingDulacrsquos criterion the nonexistence of periodic orbits InSection 4 we will study the case with risk of both socialand nonsocial contagion of obesity we prove the existenceof a unique endemic equilibrium point that is globallyasymptotically stable and also through a suitable Dulacfunction we determine the nonexistence of periodic orbitsSection 5 investigates several important aspects of our modelfrom a numerical point of view Finally in Section 6we collectsome observations and conclusions
2 Obesity Mathematical Model
The model proposed by Ejima et al [8] for the dynamicsof obesity is given by the following set of three differentialequations
119889119878 (119905)119889119905 = 120583119873 minus 120583119878 (119905) minus [120573119868 (119905) + 120598] 119878 (119905)
wherein 119878(119905) 119868(119905) and 119877(119905) respectively denote the suscep-tible (never-obese) infectious (obese) and recovered (ex-obese) individuals in a population In (1) the natural deathand birth rates are assumed to be equal and denoted by 120583thus we have 119878 + 119868 + 119877 = 119873 for all time (the populationsize is constant) Also the parameter 120573 is the transmissionrate due to social contagion risk of obesity 120574 describes therate at which the infectious individuals become recoveredindividuals 120598 is the hazard of obesity due to nonsocialcontagion reasons and 120590 is the relative risk of weight regainamong ex-obese individuals which typically takes a valuegreater than unity (120590 gt 1) due to high risk of coming backto the obese state All the involved parameters are positive
In Figure 1 the flow diagram is shown in which the modelassumptions are based and system (1) is deduced There maybe seen how it is a variant of a SIRI model since the term120590[120573119868(119905) + 120598]119877(119905) given in the second equation (1) representsthe effect of relapse that is the recovered people (ex-obese)after some time become infected (obese) again It should benoted that in total absence of the relapse term (1) is reducedto the corresponding well-known SIRmodel
Adding the three equations (1) we obtain the differentialequation for the total population of this system119889 (119878 + 119868 + 119877)119889119905 = 120583119873 minus 120583 (119878 + 119868 + 119877) (2)
It may be shown easily that the region of biological sense isgiven by
Σ = (119878 119868 119877) isin R3
+ 119878 ge 0 119868 ge 0 119877 ge 0 119878 + 119868 + 119877
= 119873 (3)
that is all solutions starting in Σ remain there for all 119905 ge 0Clearly the set Σ is positively invariant with respect to (1)
21 Reduction to a Two-Dimensional IR System We caneliminate 119878 from the equations of system (1) using the
Discrete Dynamics in Nature and Society 3
identity 119878 + 119868 + 119877 = 119873 in order to obtain the following two-dimensional system written in terms of the variables 119868119877
119889119868 (119905)119889119905 = [120573119868 (119905) + 120598] (119873 minus 119868 (119905) minus 119877 (119905))+ 120590 [120573119868 (119905) + 120598] 119877 (119905) minus (120574 + 120583) 119868 (119905) 119889119877 (119905)119889119905 = 120574119868 (119905) minus 120590 [120573119868 (119905) + 120598] 119877 (119905) minus 120583119877 (119905)
(4)
It is straightforward to prove that the region
Δ = (119868 119877) isin R2
+ 119868 ge 0 119877 ge 0 119868 + 119877 le 119873 (5)
is also a positively invariant set for the reduced model (4)With the aim to carry out a qualitative analysis of the
dynamics for system (4) we study it considering firstly thattransmission only is produced by social contagion route(120598 = 0) and afterwards taking into account both social andnonsocial contagion hazards of the obesity (120598 gt 0) In eachcasewe performed a local and global analysis of the behaviourof solutions of the system
3 Case with Only Social Contagion Risk ofObesity (120598 = 0)
31 Equilibria Local Stability and Backward Bifurcation Ifwe impose the condition 120598 = 0 in system (4) it takes thefollowing form
119889119868 (119905)119889119905 = 120573119868 (119905) (119873 minus 119868 (119905) minus 119877 (119905)) + 120590120573119868 (119905) 119877 (119905)minus (120574 + 120583) 119868 (119905) 119889119877 (119905)119889119905 = 120574119868 (119905) minus 120590120573119868 (119905) 119877 (119905) minus 120583119877 (119905)
(6)
By equalizing to zero the right members of system (6) asis commonly done to find equilibrium points we obtain thecubic polynomial equation
where the coefficients have been defined119886 equiv 1205901205732119887 equiv 120590120573 (120574 + 120583) ( 120583120574 + 120583 + 1120590 minus 1198770) 119888 equiv 120583 (120574 + 120583) (1 minus 1198770)
(8)
In (8) 1198770 is defined as
1198770 equiv 120573119873120574 + 120583 (9)
which is named as the basic reproductive number of thesystem and as is well known is a dimensionless quantitythat represents the average number of secondary infections
BifurcationBifurcationpoint
106 0804 12020
10000
20000
30000
40000
50000
60000
70000
Infe
cted
(I)
Rlowast0
Basic reproductive number (R0)
point R0 = 1
Figure 2 Schematic representation of the bifurcation diagram ofsystem (4) for the case 120598 = 0 Here is shown the variable 119868 as afunction to the parameter 1198770caused by an infective individual introduced into a group ofsusceptible individuals From the second equation (6) 119877lowast isobtained in terms of 119868lowast by means of
According to (7) and (10) we have several points ofequilibrium formodel (6) one disease-free equilibrium1198640 =(0 0) and at least two endemic equilibria 119864lowast = (119868lowast 119877lowast)Besides the solutions of (7) given by 119868lowast = 0 and
wherein 120575 equiv 1198872 minus 4119886119888 is a discriminant can be graphedas functions of 1198770 in the first quadrant of the 1198770119868 planewhere they represent different families of equilibrium pointsone formed by a line of disease-free equilibrium along thehorizontal 1198770 axis and the other two of endemic equilibriumconstituted by quadratic branches In this situation 1198770 playsthe role of a bifurcation parameter See Figure 2
In order to determine the stability of these families ofequilibrium points we need to calculate the Jacobian matrixof system (6) and evaluate it on them This matrix overall isgiven as119869 (119868 119877)
= (120573119873 minus (120574 + 120583) minus 2120573119868 + (120590 minus 1) 120573119877 (120590 minus 1) 120573119868120574 minus 120590120573119877 minus120590120573119868 minus 120583) (12)
Now we perform the local stability analysis consideringseparately the disease-free equilibrium and the endemicequilibria cases
(1) The Disease-Free Equilibrium 1198640 = (0 0) The Jacobianmatrix (12) evaluated in the disease-free equilibrium 1198640 takesthe simple form
whose eigenvalues are 1205821 = minus120583 and 1205822 = minus120583(120574 + 120583)(1 minus 1198770)Note that if 1198770 lt 1 then 1205822 lt 0 and therefore 1198640 is locallyasymptotically stable while when 1198770 gt 1 1205822 gt 0 and 1198640 isunstable In both cases the equilibrium point is hyperbolicThis situation is depicted in Figure 2 by the horizontal line onthe 1198770 axis wherein for 1198770 lt 1we have stability (indicated bya solid line) and for 1198770 gt 1 instability (showed by a dashedline) It should be noted that if 1198770 = 1 then 1205822 = 0 asa consequence the equilibrium point is nonhyperbolic and1198770 = 1 could be the value in which a bifurcation is producedHence we have the following result
Theorem 1 If 1198770 lt 1 then 1198640 is an equilibrium of system (6)and it is locally asymptotically stable Otherwise if1198770 gt 1 then1198640 is unstable(2)The Endemic Equilibria 119864lowast = (119877lowast 119868lowast) By substituting theendemic equilibria 119864lowast = (119868lowast 119877lowast) in system (6) the identitiesare obtained (120590 minus 1) 120573119877lowast = 120573119868lowast minus 120573119873 + (120574 + 120583)
120583119877lowast = 120574119868lowast minus 120590120573119868lowast119877lowast (14)
which allows writing the Jacobian matrix (12) evaluated insuch points as
119869 (119868lowast 119877lowast) = (minus120573119868lowast (120573119868lowast minus (120574 + 120583) (1 minus 1198770)) 119868lowast119877lowast120583119877lowast119868lowast minus120590120573119868lowast minus 120583 ) (15)
Det (119869) equiv 119886119868lowast2 minus 119888 (18)
are the trace and determinant of (15) respectively Becausein the Δ region both coordinates of 119864lowast are positive thenTr(119869) always is negative not so with Det(119869) which could bepositive or negative According to the discriminant 120575 of (11)in the corresponding analysis of the equilibrium points inorder to ensure that 119864lowast isin Δ the following three cases maybe identified
(i) If 120575 lt 0 then considering in this condition expres-sions 119886 119887 and 119888 given by (8) we obtain 1198770 lt 119877lowast
0
where
119877lowast0equiv (120590 minus 1) 120583 + 120574 + 2radic(120590 minus 1) 120583120574120590 (120574 + 120583) (19)
Consequently in accordance with (11) in this intervalwe have that there is not endemic equilibrium pointsSince 120583 and (120590minus1)120574 are two positive real numbers the
relation between its arithmetic and geometric meansis always given by
12 [(120590 minus 1) 120574 + 120583] ge radic(120590 minus 1) 120574120583 (20)
therefore from (19) and (20) itmay be concluded that119877lowast0le 1
(ii) If 120575 = 0 then substituting 119886 119887 and 119888 given by (8)in this equality we obtain 1198770 = 119877lowast
0 Thus according
to (11) when 1198770 reaches this value we have only oneendemic equilibrium point and 119868lowast = 119868lowast
0 where
119868lowast0equiv radic120574120583 (120590 minus 1) minus 120583120590120573 (21)
is the corresponding infected population of119877lowast0More-
over if (19) and (21) are substituted in determinant(18) and trace (17) these are simplified as
Det (119869) = 0 (22)
Tr (119869) = minus(120590 + 1)radic120574120583 (120590 minus 1) minus 120583120590 (23)
respectively Consequently taking into account (22)and (23) eigenvalues (16) are reduced to 1205821 = Tr(119869) lt0 and 1205822 = 0 one of which is negative and theother is zero Therefore this equilibrium point isnonhyperbolic and 1198770 = 119877lowast
0could also be the value
in which a bifurcation is produced See Figure 2(iii) If 120575 gt 0 then similarly using 119886 119887 and 119888 given by (8)
this inequality leads to 119877lowast0lt 1198770 As a consequence
from (11) it follows that in this interval we havefamilies of endemic equilibrium points determinedby two quadratic branches It may be shown that forall endemic equilibrium points that form the uppercurve since 119868lowast gt 119868lowast
0 from (18) it results in the fact
that Det(119869) gt 0 thereby in (16) it follows that botheigenvalues 1205821 and 1205822 are negative and thereforethese equilibriumpoints are asymptotically stable Onthe other hand it can also be shown that for endemicequilibrium points that are part of the lower curvesince 119868lowast lt 119868lowast
0 from (18) it occurs that Det(119869) lt 0
in this way again from (16) it is obtained that 1205821 gt 0and1205822 lt 0 and consequently such equilibriumpointsare unstable Note that in the latter case (in orderthat 119864lowast isin Δ) quadratic branch is delimited by thepoint (1 0) in which by the way Det(119869) = 0 andtherefore 1205821 lt 0 and 1205822 = 0 that is the said point isnonhyperbolic and in it a bifurcation could also occurThese results are illustrated in Figure 2
It is necessary to point out that from the above arguments itfollows that if 0 lt 1198770 lt 119877lowast
0 then the disease-free family of
equilibrium points 1198640 is the only one in the feasible regionΔ Based on the previous analysis we have the followingtheorem which summarizes the local stability of all endemicequilibria points of system (6) contained in Δ
Discrete Dynamics in Nature and Society 5
Theorem 2 If 0 lt 1198770 lt 119877lowast0 then there are not endemic
equilibrium points 119864lowast in the feasible region Δ Also If 1198770 =119877lowast0 then in such region there is a unique 119864lowast which is not
hyperbolic Moreover if 119877lowast0
lt 1198770 then there are familiesof endemic equilibrium points determined by two quadraticbranches an upper in which all points are asymptoticallystable and the other lower formed by unstable points and withepidemiological significance only when 1198770 ⩽ 1
In Figure 2 the results indicated by Theorems 1 and 2are displayed schematically There it may be appreciated thatthe trajectories of solutions of system (4) change abruptly inthe nonhyperbolic points (119877lowast
0 119868lowast) and (1 0) that is in these
points really bifurcations occur In that sense Figure 2 can beconsidered as the bifurcation diagram in the feasible regionΔ of said system and the type of bifurcation shown in itcorresponds to a backward bifurcation
32 Nonexistence of Periodic Orbits and Global StabilityGlobal analysis of system (6) includes the study of periodicorbits The following result shows that this system does nothave periodic orbits
Theorem 3 System (6) does not have periodic orbits in theinterior of ΔProof Consider system (6) for 119868 gt 0 and 119877 gt 0 We considerthe Dulac function
that is 120597(Φ119875)120597119868 + 120597(Φ119876)120597119877 lt 0 in the interior of Δit follows from the Dulac criterion that system (6) has noperiodic orbits in that region
On the other hand global analysis of system (6) alsoinvolves the study of global stability of its disease-freeequilibrium and its endemic equilibria The former may beproved by arguments of local stability as can be seen in thefollowing theorem
Theorem 4 If 0 lt 1198770 lt 119877lowast0 then the unique disease-free
equilibrium 1198640 of system (6) is globally asymptotically stablein ΔProof According to the discussion given previously duringthe formulation of Theorems 1 and 2 it was found that if0 lt 1198770 lt 119877lowast
0 then the disease-free equilibrium point 1198640
is the only one and it also is asymptotically stable in thefeasible region Δ Besides according to Theorem 3 there arenot periodic orbits here As a result all initial condition inΔ which satisfies this interval originates paths tending to1198640 consequently such point must be globally asymptoticallystable
Finally in order to prove the global stability of a uniqueendemic equilibrium 119864lowast in the interior of Δ when 1198770 gt 1 weuse the method of Lyapunov functions This is accomplishedin the next theorem
Theorem5 Assume that1198770 gt 1Then a unique endemic equi-librium 119864lowast = (119868lowast 119877lowast) of system (6) is globally asymptoticallystable in the interior of ΔProof To prove the global asymptotic stability of the uniqueendemic equilibrium 119864lowast we define a Lyapunov function119882(119868 119877) isin R2
0+ 119868 gt 0 119877 gt 0 rarr R given by
119882(119868 119877) = 120583(ln 119868119868lowast + 119868lowast119868 minus 1)+ 120573 (120590 minus 1) (119877 minus 119877lowast minus 119877lowast ln 119877119877lowast )
(26)
Let119882119894 = ln(119868119868lowast) + 119868lowast119868 minus 1 By using 120573119873 minus (120574 + 120583) = 120573119868lowast minus120573(120590 minus 1)119877lowast we have119889119882119894119889119905 = 120573119868lowast (2 minus 119868lowast119868 minus 119868119868lowast)
+ 120573 (120590 minus 1) 119877lowast ( 119877119877lowast minus 1 minus 119868lowast119877119868119877lowast + 119868lowast119868 ) (27)
Let119882119903 = 119877minus119877lowastminus119877lowast ln(119877119877lowast) By using 120574 = 120590120573119877lowast+120583(119877lowast119868lowast)we have
119889119882119903119889119905 = 120590120573119877lowast119868 (2 minus 119877119877lowast minus 119877lowast119877 )+ 120583119877lowast ( 119868119868lowast minus 119877119877lowast minus 119868119877lowast119868lowast119877 + 1) (28)
The derivative of (26) along solution of (6) is given by
119889119882119889119905 = 120583119889119882119894119889119905 + 120573 (120590 minus 1) 119889119882119903119889119905 (29)
By means of (27) and (28) we obtain
119889119882119889119905 = 120583120573 (119868lowast minus (120590 minus 1) 119877lowast) (2 minus 119868lowast119868 minus 119868119868lowast)+ 120573 (120590 minus 1) 120590120573119877lowast119868 (2 minus 119877119877lowast minus 119877lowast119877 )+ 120583120573 (120590 minus 1) 119877lowast (2 minus 119868119877lowast119868lowast119877 minus 119868lowast119877119868119877lowast ) lt 0
(30)
Using
120573119868lowast minus 120573 (120590 minus 1) 119877lowast = 120573119873 minus (120574 + 120583)= (120574 + 120583) (1198770 minus 1) (31)
6 Discrete Dynamics in Nature and Society
we have119889119882119889119905 = 120583 (120574 + 120583) (1198770 minus 1) (2 minus 119868lowast119868 minus 119868119868lowast)+ 120573 (120590 minus 1) 120590120573119877lowast119868 (2 minus 119877119877lowast minus 119877lowast119877 )+ 120583120573 (120590 minus 1) 119877lowast (2 minus 119868119877lowast119868lowast119877 minus 119868lowast119877119868119877lowast ) lt 0
(32)
Therefore if 1198770 gt 1 then 119889119882119889119905 is negative definite Bythe Lyapunov asymptotic stability theorem [17] this resultimplies that119864lowast is globally asymptotically stable in the interiorof ΔRemark 6 Recently Vargas-De-Leon [18] used the Lyapunovfunction119882119894 = ln(119868119868lowast)+119868lowast119868minus1 to prove global stability of thecoexistence equilibrium of two-species mutualism models
4 Case with Social and Nonsocial ContagionRisks of Obesity (120598 gt 0)
41 Equilibria and Nonexistence of Periodic Orbits Wemightfollow the usual method of setting the right-hand side of (4)equal to zero in order to obtain their equilibria This proce-dure is not the most appropriate because it leads to a cubicequation whose analytical solutions are quite complicatedInstead an alternative way to find the equilibrium points isto perform a geometric analysis based on the intersection ofthe nullclines of system (4) The nullclines = 0 (verticaldirections) and = 0 (horizontal directions) are given bythe functions
1198661 (119868) = 1205731198682 minus (120573119873 minus 120598 minus 120583 minus 120574) 119868 minus 120598119873120573 (120590 minus 1) 119868 + 120598 (120590 minus 1) (33)
respectivelyNote that1198661(119868) and1198662(119868) respectively are discontinuous
in 119868 = minus120598120573 and 119868 = minus(120590120598 + 120583)120590120573 Besides it is clear that1198661(119868) rarr +infin (resp 1198662(119868) rarr 120574120590120573) as 119868 rarr +infin and that thefunction1198661(119868) (resp1198662(119868)) is increasing and concave downGiven that 1198661(0) = minus119873(120590 minus 1) lt 0 and 1198662(0) = 0 thenthe functions 1198661(119868) and 1198662(119868) intersect at a single point (theequilibrium point) in the first quadrant (see Figure 3) Thisresult indicates that in Δ an endemic equilibrium point existsand is unique
Regarding whether or not system (4) has periodic orbitswe have precisely the following result
Theorem 7 System (4) does not have periodic orbits in theinterior of ΔProof Consider system (4) for 119868 gt 0 and 119877 gt 0 From theDulac function
Φ (119868 119877) = 1119868 (35)
Equilibrium point
minus120590120576 + 120583
120590120573minus120576
120573
minusN
120590 minus 1
120574
120590120573
R
G1(I)
G2(I)
I
Figure 3 Schematic representation of a branch of the verticalnullcline 1198661(119868) (dashed line) and the horizontal nullcline 1198662(119868)(solid line) Both curves intersect inΔ once at the equilibrium point
Equilibriumpoint
R
(0 N)
R = N minus I
I+ (N 0) I
Figure 4 Trapping region bounded by the dashed line formed bythe horizontal and vertical axes and the diagonal line 119877 = 119873 minus 119868This triangular region contains inside the only equilibrium point
it follows that120597 (Φ119875 (119868 119877))120597119868 + 120597 (Φ119876 (119868 119877))120597119877= minus 1205981198682 (119873 minus (119868 + 119877)) minus (120573 + 120598119868) minus 1205901205981198771198682minus 120590(120573 + 120598119868) minus 120583119868
(36)
Since 120597(Φ119875)120597119868+120597(Φ119876)120597119877 lt 0 inΔ taking into account theDulac criterion we conclude that system (4) has no periodicorbits
42 Local andGlobal Stability In order to determine the localand global stability of the single equilibrium point in Δ weconsider that it is inside of the triangular region bounded bythe dashed line shown in Figure 4 This is a trapping region
Discrete Dynamics in Nature and Society 7
the vector field of system (4) on the boundary point into thebox For the points on the horizontal side = 120574119868 gt 0 and = minus(119868 minus 119868+)(119868 minus 119868minus) where 119868+ and 119868minus are the positive andnegative roots of (120573119868 + 120598)(119868 minus 119873) + (120574 + 120583)119868 respectivelyThus these inequalities imply that the vector field points upand right on the interval (0 119868+) while it goes up and left on(119868+ 119873) Moreover in the vertical side = minus(120590120598 + 120583)119877 le 0and = 120598119873 + 120598(120590 minus 1)119877 gt 0 This implies that the vector fieldis directed downward and right except at the origin where itgoes to the right
Finally on the diagonal side of slope minus1 extending fromthe point (0119873) to (119873 0) it can be shown that
119889119877119889119868 = minus1 + 120583119873(119868 minus 119868lowast+) (119868 minus 119868lowast
minus) (37)
where 119868lowast+and 119868lowastminusare the positive and negative roots of 120590(120573119868 +120598)(119868 minus 119873) + (120574 + 120583)119868 respectively Hence the vector field
for large values of 119868 is almost parallel to the diagonal lineIn a more precise analysis always 119868 minus 119868lowast
minusgt 0 and 119868 minus 119868lowast
+gt0 if 119868 gt 119868lowast
+or 119868 minus 119868lowast
+lt 0 if 119868 lt 119868lowast
+ According to
(37) the first case implies that the vector field points inward(to the right) on the diagonal line because 119889119877119889119868 is morenegative than minus1 while the latter indicates that the vectorfield is directed inside (to the left) the diagonal side because119889119877119889119868 is less negative than minus1 (See Figure 4) Therefore weconclude that the triangular region is effectively a trappingregion
On the other hand Theorem 7 prohibits the existenceof closed orbits inside the triangular region Consequentlythe trajectories entering the triangular region accordingto the Poincare-Bendixson theorem [19] must converge tothe single equilibrium point located inside it Thereforethe equilibrium point inside the triangular region must beasymptotically stable Moreover since this point is the onlypoint of equilibrium in Δ it is also globally asymptoticallystable
The results of this section may be summarized in thefollowing theorem
Theorem 8 System (4) has a unique equilibrium point in thetriangular region Δ which is globally asymptotically stable
5 Numerical Results
In this section we perform a series of numerical simulationsand graphs of system (4) to illustrate the different resultsobtained for each of the two cases of interest 120598 gt 0 and120598 = 0 previously analyzed We will use the parameter valuesreported in the work developed by Ejima et al [8] theirepidemiological meanings and magnitudes are indicated inTable 1
51 Case 120598 = 0 According to the magnitudes of theparameters reported in Table 1 (19) takes the value 119877lowast
0=069363 whereas (21) acquires the value 119868lowast
0= 163304
Table 1 Baseline values and epidemiological meanings of theparameters given by Ejima et al [8]
Parameter Baseline values Epidemiological meaning
119873 100000 individuals(assumed) Population size
1120583 694 (per year) Average life expectancy at birth120573 296 times 10minus7 (peryear)
Transmission rate of obesity dueto social contagion120598 0012 (per year) Nonsocial contagion risk ofobesity120590 80 Relative hazard of obesity amongthe ex-obese1120574 358 (per year) Average duration of obesity
2000
20000
0
4000
4000
6000
600080
00
8000
1000
0
10000
1200
0
12000
1400
0
14000
1600
0
16000
1800
0
18000
2000
0
200002200024000260002800030000
2200
024
000
2600
028
000
3000
0
I (obese)
R (e
x-ob
ese)
Figure 5 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 03 obtained by considering 120573 = 127 times 10minus7Furthermore taking into consideration (10) and (11) theresulting two endemic equilibria are119864lowast
1= (20918 91378) (38)
119864lowast2= (12522 79384) (39)
which as discussed previously are asymptotically stable andunstable respectively
In Figures 5ndash9 we illustrated representative phase por-traits of the dynamic system (6) for some different values of1198770 between 0 and just over 1 according to the bifurcationdiagram shown in Figure 2 These values are obtained bymaintaining fixed in (9) 119873 120583 and 120574 but varying theparameter 120573 Thus if 1198770 takes values between 0 and 119877lowast
0
say 1198770 = 03 we have only one equilibrium point (thedisease-free equilibrium) which is globally asymptoticallystable (see Figure 5) If 1198770 = 119877lowast
0 we have two equilibrium
points one asymptotically stable disease-free equilibriumand the other endemic equilibria point which is nonhyper-bolic (see Figure 6) When 1198770 takes values between 119877lowast0 and
8 Discrete Dynamics in Nature and Society
2000
2000
0
0
4000
4000
6000
6000
8000
800010
000
10000
1200
0
12000
1400
0
14000
1600
0
16000
1800
0
18000
2000
0
20000
2200
0
2400
0I (obese)
R (e
x-ob
ese)
Figure 6 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 119877lowast0 obtained by considering 120573 = 294 times 10minus7
5000
2000
0
0
1000
0
4000
1500
0
6000
2000
0
8000
2500
0
10000
3000
0
12000
3500
0
14000
4000
0
16000
4500
0
18000
5000
0
20000
I (obese)
R (e
x-ob
ese)
Figure 7 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 08 obtained by considering 120573 = 339 times 10minus71 say 1198770 = 08 we have three equilibria one disease-free equilibrium (asymptotically stable) and two endemicequilibria (asymptotically stable and unstable) as shown inFigure 7 If 1198770 = 1 there are two equilibrium pointsone disease-free equilibrium which is nonhyperbolic and theother asymptotically stable endemic equilibrium as shown inFigure 8 Finally if 1198770 takes a slightly greater value to 1 forexample1198770 = 12 we have again two equilibrium points onedisease-free equilibrium (unstable) and the other endemicequilibrium (asymptotically stable) but both are hyperbolic(see Figure 9)
On the other hand in Figures 10ndash14 examples of theevolution in time of 119868(119905) and119877(119905) are presented for the values1198770 = 03 1198770 = 119877lowast
0 1198770 = 08 1198770 = 1 and 1198770 = 12 which
1000
00
2000
0
3000
0
4000
0
5000
0
6000
0
7000
0
8000
0
9000
0
1000
00
1100
00
1200
00
I (obese)
10000
20003000400050006000700080009000
10000110001200013000140001500016000
R (e
x-ob
ese)
Figure 8 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 1 obtained by considering 120573 = 423 times 10minus7
1000
0
10000
0
2000
0
2000
3000
0
30004000
4000
050
000
50006000
6000
07000
7000
0
80009000
8000
090
000
1000011000
1000
00
12000
1100
00
1400013000
1200
0013
0000
1400
00
1500016000
I (obese)
R (e
x-ob
ese)
Figure 9 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 12 obtained by considering 120573 = 508 times 10minus7illustrate respectively some of the dynamics shown in thephase portraits in Figures 5ndash9
52 Case 120598 gt 0 Taking into account the value 120598 =0012 gt 0 given in Table 1 in addition to the othersthere reported nullclines (33) and (34) intersect at the onlyendemic equilibrium point119864lowast = (60890 66805) (40)
which is globally asymptotically stable In Figure 15 thecorresponding phase portrait with several representativetrajectories around this equilibrium point is shown Also
Discrete Dynamics in Nature and Society 9
Time (years)
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 10 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 03 and the initial condition is(119868(0) 119877(0)) = (10000 30000)
Time (years)0 100 200 300
ObeseEx-obese
20000
15000
10000
5000
0
Figure 11 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 119877lowast
0and the initial condition is(119868(0) 119877(0)) = (20000 8000)
performing another series of numerical simulations of system(4) we show for this case in Figure 16 an example of theevolution in time of 119868(119905) and 119877(119905)6 Discussion and Conclusions
According to the results of our analysis of the Ejima et almodel we arrive at the following conclusions It has beenshown that the Ejima et al model [8] if only the socialcontagion (120598 = 0) is considered presents the two bifurcationvalues 1198770 = 119877lowast
0and 1198770 = 1 which results in what is
known as backward bifurcation From a mathematical pointof view the backward bifurcation is apparently constitutedby the combination of two different kinds of bifurcations
Time (years)
40000
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 12 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 08 and the initial condition is(119868(0) 119877(0)) = (30000 12000)
Time (years)
40000
50000
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 13 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 1 and the initial condition is(119868(0) 119877(0)) = (18000 2500)a saddle-node bifurcation that occurs at 1198770 = 119877lowast
0 whose
upper (asymptotically stable) and lower (unstable) quadraticbranches are defined for 119877lowast
0lt 1198770 and a transcritical
bifurcation that takes place in 1198770 = 1 whose horizontalbranches on the 1198770 axis are asymptotically stable for 1198770 lt 1and unstable for 1 lt 1198770 For values of 1 lt 1198770 only theupper quadratic branch asymptotically stable of the saddle-node bifurcation and the unstable horizontal branch of thetranscritical bifurcation have ameaning in the feasible regionΔ The diagram shown in Figure 2 suggests the presenceof these two bifurcations in the vicinities of the points(119877lowast0 119868lowast0) and (1 0) Of course we should go beyond the
graphic evidence and employ the qualitative analysis of localbifurcations in those points to verify our statement even
10 Discrete Dynamics in Nature and Society
Time (years)0 100 200 300
ObeseEx-obese
140000
120000
100000
80000
60000
40000
20000
0
Figure 14 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 12 and the initial condition is(119868(0) 119877(0)) = (100000 80000)
1000
0
10000
0
2000
0
2000
3000
0
30004000
4000
0
5000
0
50006000
6000
0
7000
7000
0
80009000
8000
0
9000
0
1000011000
1000
00
12000
1100
00
1400013000
1200
00
1300
00
1400
00
1500016000
I (obese)
R (e
x-ob
ese)
Figure 15 Phase portrait for system (4) for the case 120598 = 0012 gt 0(considering 120573 = 296 times 10minus7)without the realization of this depth analysis our discussionsand conclusions can be supported firmly from the resultsshown in this work of investigation
Indeed the presence of the different branches of familiesof equilibrium points shown in Figure 2 causes the system todisplay as the parameter1198770 is varied a bistable region and asa consequence the hysteresis phenomenon In order to showin more detail the presence of hysteresis in the bifurcationdiagram mentioned before suppose we start the system inthe state119860 (see Figure 17) and then slowly increase 1198770 (whichis indicated by an arrow to the right just beneath the axis1198770 of Figure 17) We remain at the origin until 1198770 = 1when the origin loses its stability At this point at the slightestdisturbance the system will jump to the state 119861 located in the
0
20000
40000
60000
80000
100000
120000
Time (years)0 100 200 300
ObeseEx-obese
Figure 16 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0012 gt 0 and the initial condition is(100000 50000)
B
A
1
C
80000
70000
60000
50000
40000
30000
20000
10000
0
Infe
cted
(I)
Rlowast0
Basic reproductive number (R0)
Figure 17 Schematic representation of hysteresis phenomenon thatexhibits the bifurcation diagram of system (4) for the case 120598 = 0Here the forward and backward paths are not equal as the parameter1198770 is changedstable upper branch Insofar as1198770 increases its value (1198770 gt 1)the equilibrium points of the system will move along thisbranch away from the state119861 If now1198770 continuously reducedits value equilibrium states will return to the state 119861 cross itand reach the state 119862 (in which 1198770 = 119877lowast
0) At this point at
the slightest change again equilibrium states will jump backto the origin (in 1198770 = 119877lowast0 ) and if 1198770 continues to decrease wewill return to the state119860 Consequently the system exhibits alack of reversibility or it is said that it has memory (or that itpresents hysteresis) because the forward and backward pathsare not identical
From an epidemiological point of view in this backwardbifurcation if 0 lt 1198770 lt 119877lowast
0 no matter where the initial
Discrete Dynamics in Nature and Society 11
conditions are taken solutions always tend to the straightline of stable disease-free points (that correspond to a never-obese population) When 119877lowast
0lt 1198770 lt 1 if the initial
conditions are located below the unstable quadratic branchof endemic equilibrium points (obese people) then solutionscontinue going to the horizontal line of stable disease-freepoints otherwise they will be directed to the stable curveof endemic points located at the upper quadratic branchFinally if 1 lt 1198770 no matter again where the initialconditions are situated solutions will continue heading forthe stable endemic points of such upper branch Should benoted that unlike what happens in systems that exhibit thephenomenon of forward bifurcation in which the endemicequilibrium exists only for 1198770 gt 1 [20] in systems exhibitinga backward bifurcation under certain initial conditions itis possible to have endemic states when 1198770 lt 1 In ourcase the endemic equilibrium that exists for 1198770 just aboveone has a large obese population so the result of 1198770 risingabove one would be a sudden and dramatic jump in thenumber of infective Moreover reducing 1198770 back below onewould not eradicate the obesity if the size of the infectedpopulation is greater or equal than 119868lowast
0 the corresponding
value of 119877lowast0 In order to eradicate the disease 1198770 must be
further reduced so it just needs to be slightly less than119877lowast0 to suddenly enter the region where only stable disease-
free equilibrium (never-obese population) exists Both suchdifferent abrupt behaviours are the result of hysteresisphenomenon
It can be seen from the bifurcation diagram shown inFigure 2 that theway inwhich obesity infected population canbe lowered (or raised) is through the reduction (or increase)of 1198770 parameter which in accordance with (9) depends onthe four quantities 119873 120573 120583 and 120574 Since we have assumedin (1) that the natural death and birth rates are equal anddenoted by 120583 the population size119873 is constantTherefore1198770can only change according to the following forms Firstly 1198770diminishes if the natural rate 120583 (of death or birth) increasesthe transmission rate 120573 decreases or the rate 120574 grows atwhich the infected individuals become recovered On theother hand 1198770 increment its value if the opposite occurs 120583decreases 120573 goes up or 120574 declines It should be noted thatchanges in 120583 could be slower and achieved in the long termprobably in decades which is not a viable option Converselychanges in 120573 or 120574 might be faster and perhaps occur infewer years Since the focus is on reducing obesity levels andtherefore the values of 1198770 it is of interest to seek mechanismsto decrease120573 andor increase 120574The former could be achievedby sensitizing at the population in general about the benefitsassociatedwith having a healthy and balanced diet and also bydesigning effective health and informative public preventionpoliciesThe latter could be done by providing better medicalassistance and making it available to the obese populationIt should be emphasized that this last strategy is not feasiblein countries with high levels of obesity as Mexico because atthe present time it is very expensive and will become evenmore expensive as the affected population increases In thisregard we believe that it is much more convenient that thegovernments of the affected countries implement broaderand more effective preventive policies
It is worth mentioning that in Mexico there have beenstudies to quantify more precisely the prevalence of peoplesuffering from overweight and obesity It is estimated that713 of Mexican adults aged 20 or more are found inthis condition [21] While trends show slowing down of theincrease in the obesity prevalence there is no evidence toinfer that prevalence will decrease in the coming years Inthis sense studies like this allow us to analyze the trends ofthe population with obesity and study the different possiblescenarios Particularly with the model studied this couldbe done for Mexico if we knew accurately the values ofits parameters 120573 and 120574 To our knowledge there is notsuch information Nevertheless these parameters could beestimated in a short or medium term based on statisti-cal information collected on obesity in our country Theknowledge of how these parameters behave over time wouldcontribute in quantifying the effectiveness of the Mexicanpublic policies against obesity
With respect to the model studied it is pertinent tocomment about the term of relapse 120590[120573119868(119905) + 120598]119877(119905) of thesecond equation (1) In the first place in this work it has beenconsidered that the parameter 120590 the relative risk of weightregain among ex-obese individuals is greater than unity (120590 gt1)This condition represents the high risk that recovered (ex-obese) individuals will become obese again However it isalso plausible to consider that 0 lt 120590 lt 1 which representsthat ex-obese population is more resistant to being obeseagain that is they are more aware of the health risks thatthis would represent The consideration that 0 lt 120590 lt 1 doesnot change the results of the stability analysis and periodicorbits performed in Sections 3 and 4 With respect to thelatter the nullcline 1198661 (given by (33)) changes to 1198661015840
1 where1198661015840
1= minus1198661 while 1198662 (34) remains the same The intersection
of the nullclines 11986610158401and 1198662 again occurs at only one point
in the region Δ and its stability is the same Regarding thisaspect we could conclude that regardless of the value ofthe parameter 120590 there is always the possibility that the ex-obese population is at risk of relapse and becoming obeseagain of course this possibility is less if 0 lt 120590 lt 1 andgreater if 120590 gt 1 In this sense public health programs againstobesity designed by governments may be more effectiveif they take into account the awareness of the ex-obesepopulation
Finally with regard to the mentioned relapse term120590[120573119868(119905) + 120598]119877(119905) if we consider only the effects of socialcontagion (which corresponds to 120598 = 0) it is reduced tothe nonlinear term 120590120573119868(119905)119877(119905) which causes system (4) toexhibit a backward bifurcation If the effects of nonsocialcontagion (that is 120598 gt 0) are also incorporated the presenceof the linear term 120590120573120598119877(119905) causes the backward bifurcationto disappear in system (4) and in its place it only presentsan equilibrium point global asymptotically stable The lattercase suggests that the way in which the effects of nonsocialcontagion are introduced into the model by means of thelinear part of the relapse term should be modified in orderto maintain much of the interesting dynamic that is presentin the backward bifurcation Certainly an alternative wayis considering instead of a linear a nonlinear contribution
12 Discrete Dynamics in Nature and Society
Perhaps it is possible to build amore general nonlinearmodelwhere in the case in which the contribution due to nonsocialeffects gradually fades the backward bifurcation of the Ejimaet al model could be obtained as a limit case
Competing Interests
The authors declare that they have no conflict of interests
Acknowledgments
This research was supported by the UACM-SECITI ResearchGrant PI-2013-34 Cruz Vargas-De-Leon would like toexpress his indebtedness to Dr Ramon Reyes Carreto for hishospitality during his stay as visiting professor at the UAGroin Chilpancingo
References
[1] World Health Organization Obesity Preventing and Managingthe Global Epidemic Report of a WHO Consultation WHOTechnical Report Series No 894 WHO Geneva Switzerland2000 httpswwwgooglecommxurlsa=tamprct=jampq=ampesrc=sampsource=webampcd=1ampved=0ahUKEwjlno 3tsDRAhVqi1QKH-W0rBPIQFggZMAAampurl=http3A2F2Fwhqlibdocwhoint2Ftrs2FWHO TRS 894pdfampusg=AFQjCNFLusNXrJz-xFKX1xpQwkHLEr-Lohgampbvm=bv144224172damcampcad=rja
[2] C L Vanvrancken-Tompkins andMaM Sothern ldquoPrevencionde la obesidad en los ninos desde el nacimiento hasta loscincoanosrdquo in Enciclopedia Sobre el Desarrollo de la PrimeraInfancia R Tremblay R Barr R Peters and M Boivin Edsvol 1ndash7 Centre of Excellence for EarlyChildhoodDevelopmentQuebec Canada 2010
[3] S K Garg H Maurer K Reed and R Selagamsetty ldquoDiabetesand cancer two diseases with obesity as a common risk factorrdquoDiabetes Obesity andMetabolism vol 16 no 2 pp 97ndash110 2014
[4] National Heart ldquoLung and Blood Institute World Health WhatAre the Health Risks of Overweight and Obesityrdquo httpswwwnhlbinihgovhealthhealth-topicstopicsoberisks
[5] World Health Organization (WHO) Global Health Observa-tory Data Repository WHO Geneva Switzerland 2011 httpappswhointghodata
[6] World Health Organization The Global Strategy on Diet Phys-ical Activity and Health (DPAS) WHO Geneva Switzerland2004 httpwwwwhointnmhwha59dpasen
[7] N A Christakis and J H Fowler ldquoThe spread of obesity in alarge social network over 32 yearsrdquoThe New England Journal ofMedicine vol 357 no 4 pp 370ndash379 2007
[8] K Ejima K Aihara and H Nishiura ldquoModeling the obesityepidemic social contagion and its implications for controlrdquoTheoretical Biology and Medical Modelling vol 10 article 17 pp1ndash13 2013
[9] A J Arenas G Gonzalez-Parra and L Jodar ldquoPeriodic solu-tions of nonautonomous differential systems modeling obesitypopulationrdquo Chaos Solitons amp Fractals vol 42 no 2 pp 1234ndash1244 2009
[10] G Gonzalez-Parra L Acedo R-J Villanueva Mico and A JArenas ldquoModeling the social obesity epidemic with stochasticnetworksrdquo Physica A Statistical Mechanics and its Applicationsvol 389 no 17 pp 3692ndash3701 2010
[11] G Gonzalez-Parra L Jodar F J Santonja and R J VillanuevaldquoAn age-structured model for childhood obesity mathematicalpopulation studiesrdquo An International Journal of MathematicalDemography vol 17 no 1 pp 1ndash11 2010
[12] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 Article ID e1000968 2010
[13] L Jodar F J Santonja and G Gonzalez-Parra ldquoModelingdynamics of infant obesity in the region of Valencia SpainrdquoComputers andMathematics with Applications vol 56 no 3 pp679ndash689 2008
[14] F-J Santonja R-J Villanueva L Jodar and G Gonzalez-Parra ldquoMathematical modelling of social obesity epidemic inthe region of Valencia Spainrdquo Mathematical and ComputerModelling of Dynamical Systems vol 16 no 1 pp 23ndash34 2010
[15] F J Santonja and L Shaikhet ldquoProbabilistic stability analysisof social obesity epidemic by a delayed stochastic modelrdquoNonlinear Analysis Real World Applications vol 17 pp 114ndash1252014
[16] D Tudor ldquoA deterministic model for herpes infections inhuman and animal populationsrdquo SIAM Review vol 32 no 1pp 136ndash139 1990
[17] A M Lyapunov The general problem of the stability of motionTaylor amp Francis London England 1992
[18] C Vargas-De-Leon ldquoLyapunov functions for two-species coop-erative systemsrdquo Applied Mathematics and Computation vol219 no 5 pp 2493ndash2497 2012
[19] D W Jordan and P Smith Nonlinear Ordinary DifferentialEquations An Introduction for Scientists and Engineers OxfordUniversity Press New York NY USA 4th edition 2007
[20] C M Kribs-Zaleta and J X Velasco-Hernandez ldquoA simplevaccination model with multiple endemic statesrdquoMathematicalBiosciences vol 164 no 2 pp 183ndash201 2000
[21] S Barquera I Campos-Nonato L Hernandez-Barrera APedroza-Tobıas and J A Rivera-Dommarco ldquoPrevalence ofobesity in mexican adults ensanut 2012rdquo Salud Publica deMexico vol 55 supplement 2 pp S151ndashS160 2013
identity 119878 + 119868 + 119877 = 119873 in order to obtain the following two-dimensional system written in terms of the variables 119868119877
119889119868 (119905)119889119905 = [120573119868 (119905) + 120598] (119873 minus 119868 (119905) minus 119877 (119905))+ 120590 [120573119868 (119905) + 120598] 119877 (119905) minus (120574 + 120583) 119868 (119905) 119889119877 (119905)119889119905 = 120574119868 (119905) minus 120590 [120573119868 (119905) + 120598] 119877 (119905) minus 120583119877 (119905)
(4)
It is straightforward to prove that the region
Δ = (119868 119877) isin R2
+ 119868 ge 0 119877 ge 0 119868 + 119877 le 119873 (5)
is also a positively invariant set for the reduced model (4)With the aim to carry out a qualitative analysis of the
dynamics for system (4) we study it considering firstly thattransmission only is produced by social contagion route(120598 = 0) and afterwards taking into account both social andnonsocial contagion hazards of the obesity (120598 gt 0) In eachcasewe performed a local and global analysis of the behaviourof solutions of the system
3 Case with Only Social Contagion Risk ofObesity (120598 = 0)
31 Equilibria Local Stability and Backward Bifurcation Ifwe impose the condition 120598 = 0 in system (4) it takes thefollowing form
119889119868 (119905)119889119905 = 120573119868 (119905) (119873 minus 119868 (119905) minus 119877 (119905)) + 120590120573119868 (119905) 119877 (119905)minus (120574 + 120583) 119868 (119905) 119889119877 (119905)119889119905 = 120574119868 (119905) minus 120590120573119868 (119905) 119877 (119905) minus 120583119877 (119905)
(6)
By equalizing to zero the right members of system (6) asis commonly done to find equilibrium points we obtain thecubic polynomial equation
where the coefficients have been defined119886 equiv 1205901205732119887 equiv 120590120573 (120574 + 120583) ( 120583120574 + 120583 + 1120590 minus 1198770) 119888 equiv 120583 (120574 + 120583) (1 minus 1198770)
(8)
In (8) 1198770 is defined as
1198770 equiv 120573119873120574 + 120583 (9)
which is named as the basic reproductive number of thesystem and as is well known is a dimensionless quantitythat represents the average number of secondary infections
BifurcationBifurcationpoint
106 0804 12020
10000
20000
30000
40000
50000
60000
70000
Infe
cted
(I)
Rlowast0
Basic reproductive number (R0)
point R0 = 1
Figure 2 Schematic representation of the bifurcation diagram ofsystem (4) for the case 120598 = 0 Here is shown the variable 119868 as afunction to the parameter 1198770caused by an infective individual introduced into a group ofsusceptible individuals From the second equation (6) 119877lowast isobtained in terms of 119868lowast by means of
According to (7) and (10) we have several points ofequilibrium formodel (6) one disease-free equilibrium1198640 =(0 0) and at least two endemic equilibria 119864lowast = (119868lowast 119877lowast)Besides the solutions of (7) given by 119868lowast = 0 and
wherein 120575 equiv 1198872 minus 4119886119888 is a discriminant can be graphedas functions of 1198770 in the first quadrant of the 1198770119868 planewhere they represent different families of equilibrium pointsone formed by a line of disease-free equilibrium along thehorizontal 1198770 axis and the other two of endemic equilibriumconstituted by quadratic branches In this situation 1198770 playsthe role of a bifurcation parameter See Figure 2
In order to determine the stability of these families ofequilibrium points we need to calculate the Jacobian matrixof system (6) and evaluate it on them This matrix overall isgiven as119869 (119868 119877)
= (120573119873 minus (120574 + 120583) minus 2120573119868 + (120590 minus 1) 120573119877 (120590 minus 1) 120573119868120574 minus 120590120573119877 minus120590120573119868 minus 120583) (12)
Now we perform the local stability analysis consideringseparately the disease-free equilibrium and the endemicequilibria cases
(1) The Disease-Free Equilibrium 1198640 = (0 0) The Jacobianmatrix (12) evaluated in the disease-free equilibrium 1198640 takesthe simple form
whose eigenvalues are 1205821 = minus120583 and 1205822 = minus120583(120574 + 120583)(1 minus 1198770)Note that if 1198770 lt 1 then 1205822 lt 0 and therefore 1198640 is locallyasymptotically stable while when 1198770 gt 1 1205822 gt 0 and 1198640 isunstable In both cases the equilibrium point is hyperbolicThis situation is depicted in Figure 2 by the horizontal line onthe 1198770 axis wherein for 1198770 lt 1we have stability (indicated bya solid line) and for 1198770 gt 1 instability (showed by a dashedline) It should be noted that if 1198770 = 1 then 1205822 = 0 asa consequence the equilibrium point is nonhyperbolic and1198770 = 1 could be the value in which a bifurcation is producedHence we have the following result
Theorem 1 If 1198770 lt 1 then 1198640 is an equilibrium of system (6)and it is locally asymptotically stable Otherwise if1198770 gt 1 then1198640 is unstable(2)The Endemic Equilibria 119864lowast = (119877lowast 119868lowast) By substituting theendemic equilibria 119864lowast = (119868lowast 119877lowast) in system (6) the identitiesare obtained (120590 minus 1) 120573119877lowast = 120573119868lowast minus 120573119873 + (120574 + 120583)
120583119877lowast = 120574119868lowast minus 120590120573119868lowast119877lowast (14)
which allows writing the Jacobian matrix (12) evaluated insuch points as
119869 (119868lowast 119877lowast) = (minus120573119868lowast (120573119868lowast minus (120574 + 120583) (1 minus 1198770)) 119868lowast119877lowast120583119877lowast119868lowast minus120590120573119868lowast minus 120583 ) (15)
Det (119869) equiv 119886119868lowast2 minus 119888 (18)
are the trace and determinant of (15) respectively Becausein the Δ region both coordinates of 119864lowast are positive thenTr(119869) always is negative not so with Det(119869) which could bepositive or negative According to the discriminant 120575 of (11)in the corresponding analysis of the equilibrium points inorder to ensure that 119864lowast isin Δ the following three cases maybe identified
(i) If 120575 lt 0 then considering in this condition expres-sions 119886 119887 and 119888 given by (8) we obtain 1198770 lt 119877lowast
0
where
119877lowast0equiv (120590 minus 1) 120583 + 120574 + 2radic(120590 minus 1) 120583120574120590 (120574 + 120583) (19)
Consequently in accordance with (11) in this intervalwe have that there is not endemic equilibrium pointsSince 120583 and (120590minus1)120574 are two positive real numbers the
relation between its arithmetic and geometric meansis always given by
12 [(120590 minus 1) 120574 + 120583] ge radic(120590 minus 1) 120574120583 (20)
therefore from (19) and (20) itmay be concluded that119877lowast0le 1
(ii) If 120575 = 0 then substituting 119886 119887 and 119888 given by (8)in this equality we obtain 1198770 = 119877lowast
0 Thus according
to (11) when 1198770 reaches this value we have only oneendemic equilibrium point and 119868lowast = 119868lowast
0 where
119868lowast0equiv radic120574120583 (120590 minus 1) minus 120583120590120573 (21)
is the corresponding infected population of119877lowast0More-
over if (19) and (21) are substituted in determinant(18) and trace (17) these are simplified as
Det (119869) = 0 (22)
Tr (119869) = minus(120590 + 1)radic120574120583 (120590 minus 1) minus 120583120590 (23)
respectively Consequently taking into account (22)and (23) eigenvalues (16) are reduced to 1205821 = Tr(119869) lt0 and 1205822 = 0 one of which is negative and theother is zero Therefore this equilibrium point isnonhyperbolic and 1198770 = 119877lowast
0could also be the value
in which a bifurcation is produced See Figure 2(iii) If 120575 gt 0 then similarly using 119886 119887 and 119888 given by (8)
this inequality leads to 119877lowast0lt 1198770 As a consequence
from (11) it follows that in this interval we havefamilies of endemic equilibrium points determinedby two quadratic branches It may be shown that forall endemic equilibrium points that form the uppercurve since 119868lowast gt 119868lowast
0 from (18) it results in the fact
that Det(119869) gt 0 thereby in (16) it follows that botheigenvalues 1205821 and 1205822 are negative and thereforethese equilibriumpoints are asymptotically stable Onthe other hand it can also be shown that for endemicequilibrium points that are part of the lower curvesince 119868lowast lt 119868lowast
0 from (18) it occurs that Det(119869) lt 0
in this way again from (16) it is obtained that 1205821 gt 0and1205822 lt 0 and consequently such equilibriumpointsare unstable Note that in the latter case (in orderthat 119864lowast isin Δ) quadratic branch is delimited by thepoint (1 0) in which by the way Det(119869) = 0 andtherefore 1205821 lt 0 and 1205822 = 0 that is the said point isnonhyperbolic and in it a bifurcation could also occurThese results are illustrated in Figure 2
It is necessary to point out that from the above arguments itfollows that if 0 lt 1198770 lt 119877lowast
0 then the disease-free family of
equilibrium points 1198640 is the only one in the feasible regionΔ Based on the previous analysis we have the followingtheorem which summarizes the local stability of all endemicequilibria points of system (6) contained in Δ
Discrete Dynamics in Nature and Society 5
Theorem 2 If 0 lt 1198770 lt 119877lowast0 then there are not endemic
equilibrium points 119864lowast in the feasible region Δ Also If 1198770 =119877lowast0 then in such region there is a unique 119864lowast which is not
hyperbolic Moreover if 119877lowast0
lt 1198770 then there are familiesof endemic equilibrium points determined by two quadraticbranches an upper in which all points are asymptoticallystable and the other lower formed by unstable points and withepidemiological significance only when 1198770 ⩽ 1
In Figure 2 the results indicated by Theorems 1 and 2are displayed schematically There it may be appreciated thatthe trajectories of solutions of system (4) change abruptly inthe nonhyperbolic points (119877lowast
0 119868lowast) and (1 0) that is in these
points really bifurcations occur In that sense Figure 2 can beconsidered as the bifurcation diagram in the feasible regionΔ of said system and the type of bifurcation shown in itcorresponds to a backward bifurcation
32 Nonexistence of Periodic Orbits and Global StabilityGlobal analysis of system (6) includes the study of periodicorbits The following result shows that this system does nothave periodic orbits
Theorem 3 System (6) does not have periodic orbits in theinterior of ΔProof Consider system (6) for 119868 gt 0 and 119877 gt 0 We considerthe Dulac function
that is 120597(Φ119875)120597119868 + 120597(Φ119876)120597119877 lt 0 in the interior of Δit follows from the Dulac criterion that system (6) has noperiodic orbits in that region
On the other hand global analysis of system (6) alsoinvolves the study of global stability of its disease-freeequilibrium and its endemic equilibria The former may beproved by arguments of local stability as can be seen in thefollowing theorem
Theorem 4 If 0 lt 1198770 lt 119877lowast0 then the unique disease-free
equilibrium 1198640 of system (6) is globally asymptotically stablein ΔProof According to the discussion given previously duringthe formulation of Theorems 1 and 2 it was found that if0 lt 1198770 lt 119877lowast
0 then the disease-free equilibrium point 1198640
is the only one and it also is asymptotically stable in thefeasible region Δ Besides according to Theorem 3 there arenot periodic orbits here As a result all initial condition inΔ which satisfies this interval originates paths tending to1198640 consequently such point must be globally asymptoticallystable
Finally in order to prove the global stability of a uniqueendemic equilibrium 119864lowast in the interior of Δ when 1198770 gt 1 weuse the method of Lyapunov functions This is accomplishedin the next theorem
Theorem5 Assume that1198770 gt 1Then a unique endemic equi-librium 119864lowast = (119868lowast 119877lowast) of system (6) is globally asymptoticallystable in the interior of ΔProof To prove the global asymptotic stability of the uniqueendemic equilibrium 119864lowast we define a Lyapunov function119882(119868 119877) isin R2
0+ 119868 gt 0 119877 gt 0 rarr R given by
119882(119868 119877) = 120583(ln 119868119868lowast + 119868lowast119868 minus 1)+ 120573 (120590 minus 1) (119877 minus 119877lowast minus 119877lowast ln 119877119877lowast )
(26)
Let119882119894 = ln(119868119868lowast) + 119868lowast119868 minus 1 By using 120573119873 minus (120574 + 120583) = 120573119868lowast minus120573(120590 minus 1)119877lowast we have119889119882119894119889119905 = 120573119868lowast (2 minus 119868lowast119868 minus 119868119868lowast)
+ 120573 (120590 minus 1) 119877lowast ( 119877119877lowast minus 1 minus 119868lowast119877119868119877lowast + 119868lowast119868 ) (27)
Let119882119903 = 119877minus119877lowastminus119877lowast ln(119877119877lowast) By using 120574 = 120590120573119877lowast+120583(119877lowast119868lowast)we have
119889119882119903119889119905 = 120590120573119877lowast119868 (2 minus 119877119877lowast minus 119877lowast119877 )+ 120583119877lowast ( 119868119868lowast minus 119877119877lowast minus 119868119877lowast119868lowast119877 + 1) (28)
The derivative of (26) along solution of (6) is given by
119889119882119889119905 = 120583119889119882119894119889119905 + 120573 (120590 minus 1) 119889119882119903119889119905 (29)
By means of (27) and (28) we obtain
119889119882119889119905 = 120583120573 (119868lowast minus (120590 minus 1) 119877lowast) (2 minus 119868lowast119868 minus 119868119868lowast)+ 120573 (120590 minus 1) 120590120573119877lowast119868 (2 minus 119877119877lowast minus 119877lowast119877 )+ 120583120573 (120590 minus 1) 119877lowast (2 minus 119868119877lowast119868lowast119877 minus 119868lowast119877119868119877lowast ) lt 0
(30)
Using
120573119868lowast minus 120573 (120590 minus 1) 119877lowast = 120573119873 minus (120574 + 120583)= (120574 + 120583) (1198770 minus 1) (31)
6 Discrete Dynamics in Nature and Society
we have119889119882119889119905 = 120583 (120574 + 120583) (1198770 minus 1) (2 minus 119868lowast119868 minus 119868119868lowast)+ 120573 (120590 minus 1) 120590120573119877lowast119868 (2 minus 119877119877lowast minus 119877lowast119877 )+ 120583120573 (120590 minus 1) 119877lowast (2 minus 119868119877lowast119868lowast119877 minus 119868lowast119877119868119877lowast ) lt 0
(32)
Therefore if 1198770 gt 1 then 119889119882119889119905 is negative definite Bythe Lyapunov asymptotic stability theorem [17] this resultimplies that119864lowast is globally asymptotically stable in the interiorof ΔRemark 6 Recently Vargas-De-Leon [18] used the Lyapunovfunction119882119894 = ln(119868119868lowast)+119868lowast119868minus1 to prove global stability of thecoexistence equilibrium of two-species mutualism models
4 Case with Social and Nonsocial ContagionRisks of Obesity (120598 gt 0)
41 Equilibria and Nonexistence of Periodic Orbits Wemightfollow the usual method of setting the right-hand side of (4)equal to zero in order to obtain their equilibria This proce-dure is not the most appropriate because it leads to a cubicequation whose analytical solutions are quite complicatedInstead an alternative way to find the equilibrium points isto perform a geometric analysis based on the intersection ofthe nullclines of system (4) The nullclines = 0 (verticaldirections) and = 0 (horizontal directions) are given bythe functions
1198661 (119868) = 1205731198682 minus (120573119873 minus 120598 minus 120583 minus 120574) 119868 minus 120598119873120573 (120590 minus 1) 119868 + 120598 (120590 minus 1) (33)
respectivelyNote that1198661(119868) and1198662(119868) respectively are discontinuous
in 119868 = minus120598120573 and 119868 = minus(120590120598 + 120583)120590120573 Besides it is clear that1198661(119868) rarr +infin (resp 1198662(119868) rarr 120574120590120573) as 119868 rarr +infin and that thefunction1198661(119868) (resp1198662(119868)) is increasing and concave downGiven that 1198661(0) = minus119873(120590 minus 1) lt 0 and 1198662(0) = 0 thenthe functions 1198661(119868) and 1198662(119868) intersect at a single point (theequilibrium point) in the first quadrant (see Figure 3) Thisresult indicates that in Δ an endemic equilibrium point existsand is unique
Regarding whether or not system (4) has periodic orbitswe have precisely the following result
Theorem 7 System (4) does not have periodic orbits in theinterior of ΔProof Consider system (4) for 119868 gt 0 and 119877 gt 0 From theDulac function
Φ (119868 119877) = 1119868 (35)
Equilibrium point
minus120590120576 + 120583
120590120573minus120576
120573
minusN
120590 minus 1
120574
120590120573
R
G1(I)
G2(I)
I
Figure 3 Schematic representation of a branch of the verticalnullcline 1198661(119868) (dashed line) and the horizontal nullcline 1198662(119868)(solid line) Both curves intersect inΔ once at the equilibrium point
Equilibriumpoint
R
(0 N)
R = N minus I
I+ (N 0) I
Figure 4 Trapping region bounded by the dashed line formed bythe horizontal and vertical axes and the diagonal line 119877 = 119873 minus 119868This triangular region contains inside the only equilibrium point
it follows that120597 (Φ119875 (119868 119877))120597119868 + 120597 (Φ119876 (119868 119877))120597119877= minus 1205981198682 (119873 minus (119868 + 119877)) minus (120573 + 120598119868) minus 1205901205981198771198682minus 120590(120573 + 120598119868) minus 120583119868
(36)
Since 120597(Φ119875)120597119868+120597(Φ119876)120597119877 lt 0 inΔ taking into account theDulac criterion we conclude that system (4) has no periodicorbits
42 Local andGlobal Stability In order to determine the localand global stability of the single equilibrium point in Δ weconsider that it is inside of the triangular region bounded bythe dashed line shown in Figure 4 This is a trapping region
Discrete Dynamics in Nature and Society 7
the vector field of system (4) on the boundary point into thebox For the points on the horizontal side = 120574119868 gt 0 and = minus(119868 minus 119868+)(119868 minus 119868minus) where 119868+ and 119868minus are the positive andnegative roots of (120573119868 + 120598)(119868 minus 119873) + (120574 + 120583)119868 respectivelyThus these inequalities imply that the vector field points upand right on the interval (0 119868+) while it goes up and left on(119868+ 119873) Moreover in the vertical side = minus(120590120598 + 120583)119877 le 0and = 120598119873 + 120598(120590 minus 1)119877 gt 0 This implies that the vector fieldis directed downward and right except at the origin where itgoes to the right
Finally on the diagonal side of slope minus1 extending fromthe point (0119873) to (119873 0) it can be shown that
119889119877119889119868 = minus1 + 120583119873(119868 minus 119868lowast+) (119868 minus 119868lowast
minus) (37)
where 119868lowast+and 119868lowastminusare the positive and negative roots of 120590(120573119868 +120598)(119868 minus 119873) + (120574 + 120583)119868 respectively Hence the vector field
for large values of 119868 is almost parallel to the diagonal lineIn a more precise analysis always 119868 minus 119868lowast
minusgt 0 and 119868 minus 119868lowast
+gt0 if 119868 gt 119868lowast
+or 119868 minus 119868lowast
+lt 0 if 119868 lt 119868lowast
+ According to
(37) the first case implies that the vector field points inward(to the right) on the diagonal line because 119889119877119889119868 is morenegative than minus1 while the latter indicates that the vectorfield is directed inside (to the left) the diagonal side because119889119877119889119868 is less negative than minus1 (See Figure 4) Therefore weconclude that the triangular region is effectively a trappingregion
On the other hand Theorem 7 prohibits the existenceof closed orbits inside the triangular region Consequentlythe trajectories entering the triangular region accordingto the Poincare-Bendixson theorem [19] must converge tothe single equilibrium point located inside it Thereforethe equilibrium point inside the triangular region must beasymptotically stable Moreover since this point is the onlypoint of equilibrium in Δ it is also globally asymptoticallystable
The results of this section may be summarized in thefollowing theorem
Theorem 8 System (4) has a unique equilibrium point in thetriangular region Δ which is globally asymptotically stable
5 Numerical Results
In this section we perform a series of numerical simulationsand graphs of system (4) to illustrate the different resultsobtained for each of the two cases of interest 120598 gt 0 and120598 = 0 previously analyzed We will use the parameter valuesreported in the work developed by Ejima et al [8] theirepidemiological meanings and magnitudes are indicated inTable 1
51 Case 120598 = 0 According to the magnitudes of theparameters reported in Table 1 (19) takes the value 119877lowast
0=069363 whereas (21) acquires the value 119868lowast
0= 163304
Table 1 Baseline values and epidemiological meanings of theparameters given by Ejima et al [8]
Parameter Baseline values Epidemiological meaning
119873 100000 individuals(assumed) Population size
1120583 694 (per year) Average life expectancy at birth120573 296 times 10minus7 (peryear)
Transmission rate of obesity dueto social contagion120598 0012 (per year) Nonsocial contagion risk ofobesity120590 80 Relative hazard of obesity amongthe ex-obese1120574 358 (per year) Average duration of obesity
2000
20000
0
4000
4000
6000
600080
00
8000
1000
0
10000
1200
0
12000
1400
0
14000
1600
0
16000
1800
0
18000
2000
0
200002200024000260002800030000
2200
024
000
2600
028
000
3000
0
I (obese)
R (e
x-ob
ese)
Figure 5 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 03 obtained by considering 120573 = 127 times 10minus7Furthermore taking into consideration (10) and (11) theresulting two endemic equilibria are119864lowast
1= (20918 91378) (38)
119864lowast2= (12522 79384) (39)
which as discussed previously are asymptotically stable andunstable respectively
In Figures 5ndash9 we illustrated representative phase por-traits of the dynamic system (6) for some different values of1198770 between 0 and just over 1 according to the bifurcationdiagram shown in Figure 2 These values are obtained bymaintaining fixed in (9) 119873 120583 and 120574 but varying theparameter 120573 Thus if 1198770 takes values between 0 and 119877lowast
0
say 1198770 = 03 we have only one equilibrium point (thedisease-free equilibrium) which is globally asymptoticallystable (see Figure 5) If 1198770 = 119877lowast
0 we have two equilibrium
points one asymptotically stable disease-free equilibriumand the other endemic equilibria point which is nonhyper-bolic (see Figure 6) When 1198770 takes values between 119877lowast0 and
8 Discrete Dynamics in Nature and Society
2000
2000
0
0
4000
4000
6000
6000
8000
800010
000
10000
1200
0
12000
1400
0
14000
1600
0
16000
1800
0
18000
2000
0
20000
2200
0
2400
0I (obese)
R (e
x-ob
ese)
Figure 6 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 119877lowast0 obtained by considering 120573 = 294 times 10minus7
5000
2000
0
0
1000
0
4000
1500
0
6000
2000
0
8000
2500
0
10000
3000
0
12000
3500
0
14000
4000
0
16000
4500
0
18000
5000
0
20000
I (obese)
R (e
x-ob
ese)
Figure 7 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 08 obtained by considering 120573 = 339 times 10minus71 say 1198770 = 08 we have three equilibria one disease-free equilibrium (asymptotically stable) and two endemicequilibria (asymptotically stable and unstable) as shown inFigure 7 If 1198770 = 1 there are two equilibrium pointsone disease-free equilibrium which is nonhyperbolic and theother asymptotically stable endemic equilibrium as shown inFigure 8 Finally if 1198770 takes a slightly greater value to 1 forexample1198770 = 12 we have again two equilibrium points onedisease-free equilibrium (unstable) and the other endemicequilibrium (asymptotically stable) but both are hyperbolic(see Figure 9)
On the other hand in Figures 10ndash14 examples of theevolution in time of 119868(119905) and119877(119905) are presented for the values1198770 = 03 1198770 = 119877lowast
0 1198770 = 08 1198770 = 1 and 1198770 = 12 which
1000
00
2000
0
3000
0
4000
0
5000
0
6000
0
7000
0
8000
0
9000
0
1000
00
1100
00
1200
00
I (obese)
10000
20003000400050006000700080009000
10000110001200013000140001500016000
R (e
x-ob
ese)
Figure 8 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 1 obtained by considering 120573 = 423 times 10minus7
1000
0
10000
0
2000
0
2000
3000
0
30004000
4000
050
000
50006000
6000
07000
7000
0
80009000
8000
090
000
1000011000
1000
00
12000
1100
00
1400013000
1200
0013
0000
1400
00
1500016000
I (obese)
R (e
x-ob
ese)
Figure 9 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 12 obtained by considering 120573 = 508 times 10minus7illustrate respectively some of the dynamics shown in thephase portraits in Figures 5ndash9
52 Case 120598 gt 0 Taking into account the value 120598 =0012 gt 0 given in Table 1 in addition to the othersthere reported nullclines (33) and (34) intersect at the onlyendemic equilibrium point119864lowast = (60890 66805) (40)
which is globally asymptotically stable In Figure 15 thecorresponding phase portrait with several representativetrajectories around this equilibrium point is shown Also
Discrete Dynamics in Nature and Society 9
Time (years)
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 10 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 03 and the initial condition is(119868(0) 119877(0)) = (10000 30000)
Time (years)0 100 200 300
ObeseEx-obese
20000
15000
10000
5000
0
Figure 11 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 119877lowast
0and the initial condition is(119868(0) 119877(0)) = (20000 8000)
performing another series of numerical simulations of system(4) we show for this case in Figure 16 an example of theevolution in time of 119868(119905) and 119877(119905)6 Discussion and Conclusions
According to the results of our analysis of the Ejima et almodel we arrive at the following conclusions It has beenshown that the Ejima et al model [8] if only the socialcontagion (120598 = 0) is considered presents the two bifurcationvalues 1198770 = 119877lowast
0and 1198770 = 1 which results in what is
known as backward bifurcation From a mathematical pointof view the backward bifurcation is apparently constitutedby the combination of two different kinds of bifurcations
Time (years)
40000
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 12 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 08 and the initial condition is(119868(0) 119877(0)) = (30000 12000)
Time (years)
40000
50000
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 13 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 1 and the initial condition is(119868(0) 119877(0)) = (18000 2500)a saddle-node bifurcation that occurs at 1198770 = 119877lowast
0 whose
upper (asymptotically stable) and lower (unstable) quadraticbranches are defined for 119877lowast
0lt 1198770 and a transcritical
bifurcation that takes place in 1198770 = 1 whose horizontalbranches on the 1198770 axis are asymptotically stable for 1198770 lt 1and unstable for 1 lt 1198770 For values of 1 lt 1198770 only theupper quadratic branch asymptotically stable of the saddle-node bifurcation and the unstable horizontal branch of thetranscritical bifurcation have ameaning in the feasible regionΔ The diagram shown in Figure 2 suggests the presenceof these two bifurcations in the vicinities of the points(119877lowast0 119868lowast0) and (1 0) Of course we should go beyond the
graphic evidence and employ the qualitative analysis of localbifurcations in those points to verify our statement even
10 Discrete Dynamics in Nature and Society
Time (years)0 100 200 300
ObeseEx-obese
140000
120000
100000
80000
60000
40000
20000
0
Figure 14 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 12 and the initial condition is(119868(0) 119877(0)) = (100000 80000)
1000
0
10000
0
2000
0
2000
3000
0
30004000
4000
0
5000
0
50006000
6000
0
7000
7000
0
80009000
8000
0
9000
0
1000011000
1000
00
12000
1100
00
1400013000
1200
00
1300
00
1400
00
1500016000
I (obese)
R (e
x-ob
ese)
Figure 15 Phase portrait for system (4) for the case 120598 = 0012 gt 0(considering 120573 = 296 times 10minus7)without the realization of this depth analysis our discussionsand conclusions can be supported firmly from the resultsshown in this work of investigation
Indeed the presence of the different branches of familiesof equilibrium points shown in Figure 2 causes the system todisplay as the parameter1198770 is varied a bistable region and asa consequence the hysteresis phenomenon In order to showin more detail the presence of hysteresis in the bifurcationdiagram mentioned before suppose we start the system inthe state119860 (see Figure 17) and then slowly increase 1198770 (whichis indicated by an arrow to the right just beneath the axis1198770 of Figure 17) We remain at the origin until 1198770 = 1when the origin loses its stability At this point at the slightestdisturbance the system will jump to the state 119861 located in the
0
20000
40000
60000
80000
100000
120000
Time (years)0 100 200 300
ObeseEx-obese
Figure 16 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0012 gt 0 and the initial condition is(100000 50000)
B
A
1
C
80000
70000
60000
50000
40000
30000
20000
10000
0
Infe
cted
(I)
Rlowast0
Basic reproductive number (R0)
Figure 17 Schematic representation of hysteresis phenomenon thatexhibits the bifurcation diagram of system (4) for the case 120598 = 0Here the forward and backward paths are not equal as the parameter1198770 is changedstable upper branch Insofar as1198770 increases its value (1198770 gt 1)the equilibrium points of the system will move along thisbranch away from the state119861 If now1198770 continuously reducedits value equilibrium states will return to the state 119861 cross itand reach the state 119862 (in which 1198770 = 119877lowast
0) At this point at
the slightest change again equilibrium states will jump backto the origin (in 1198770 = 119877lowast0 ) and if 1198770 continues to decrease wewill return to the state119860 Consequently the system exhibits alack of reversibility or it is said that it has memory (or that itpresents hysteresis) because the forward and backward pathsare not identical
From an epidemiological point of view in this backwardbifurcation if 0 lt 1198770 lt 119877lowast
0 no matter where the initial
Discrete Dynamics in Nature and Society 11
conditions are taken solutions always tend to the straightline of stable disease-free points (that correspond to a never-obese population) When 119877lowast
0lt 1198770 lt 1 if the initial
conditions are located below the unstable quadratic branchof endemic equilibrium points (obese people) then solutionscontinue going to the horizontal line of stable disease-freepoints otherwise they will be directed to the stable curveof endemic points located at the upper quadratic branchFinally if 1 lt 1198770 no matter again where the initialconditions are situated solutions will continue heading forthe stable endemic points of such upper branch Should benoted that unlike what happens in systems that exhibit thephenomenon of forward bifurcation in which the endemicequilibrium exists only for 1198770 gt 1 [20] in systems exhibitinga backward bifurcation under certain initial conditions itis possible to have endemic states when 1198770 lt 1 In ourcase the endemic equilibrium that exists for 1198770 just aboveone has a large obese population so the result of 1198770 risingabove one would be a sudden and dramatic jump in thenumber of infective Moreover reducing 1198770 back below onewould not eradicate the obesity if the size of the infectedpopulation is greater or equal than 119868lowast
0 the corresponding
value of 119877lowast0 In order to eradicate the disease 1198770 must be
further reduced so it just needs to be slightly less than119877lowast0 to suddenly enter the region where only stable disease-
free equilibrium (never-obese population) exists Both suchdifferent abrupt behaviours are the result of hysteresisphenomenon
It can be seen from the bifurcation diagram shown inFigure 2 that theway inwhich obesity infected population canbe lowered (or raised) is through the reduction (or increase)of 1198770 parameter which in accordance with (9) depends onthe four quantities 119873 120573 120583 and 120574 Since we have assumedin (1) that the natural death and birth rates are equal anddenoted by 120583 the population size119873 is constantTherefore1198770can only change according to the following forms Firstly 1198770diminishes if the natural rate 120583 (of death or birth) increasesthe transmission rate 120573 decreases or the rate 120574 grows atwhich the infected individuals become recovered On theother hand 1198770 increment its value if the opposite occurs 120583decreases 120573 goes up or 120574 declines It should be noted thatchanges in 120583 could be slower and achieved in the long termprobably in decades which is not a viable option Converselychanges in 120573 or 120574 might be faster and perhaps occur infewer years Since the focus is on reducing obesity levels andtherefore the values of 1198770 it is of interest to seek mechanismsto decrease120573 andor increase 120574The former could be achievedby sensitizing at the population in general about the benefitsassociatedwith having a healthy and balanced diet and also bydesigning effective health and informative public preventionpoliciesThe latter could be done by providing better medicalassistance and making it available to the obese populationIt should be emphasized that this last strategy is not feasiblein countries with high levels of obesity as Mexico because atthe present time it is very expensive and will become evenmore expensive as the affected population increases In thisregard we believe that it is much more convenient that thegovernments of the affected countries implement broaderand more effective preventive policies
It is worth mentioning that in Mexico there have beenstudies to quantify more precisely the prevalence of peoplesuffering from overweight and obesity It is estimated that713 of Mexican adults aged 20 or more are found inthis condition [21] While trends show slowing down of theincrease in the obesity prevalence there is no evidence toinfer that prevalence will decrease in the coming years Inthis sense studies like this allow us to analyze the trends ofthe population with obesity and study the different possiblescenarios Particularly with the model studied this couldbe done for Mexico if we knew accurately the values ofits parameters 120573 and 120574 To our knowledge there is notsuch information Nevertheless these parameters could beestimated in a short or medium term based on statisti-cal information collected on obesity in our country Theknowledge of how these parameters behave over time wouldcontribute in quantifying the effectiveness of the Mexicanpublic policies against obesity
With respect to the model studied it is pertinent tocomment about the term of relapse 120590[120573119868(119905) + 120598]119877(119905) of thesecond equation (1) In the first place in this work it has beenconsidered that the parameter 120590 the relative risk of weightregain among ex-obese individuals is greater than unity (120590 gt1)This condition represents the high risk that recovered (ex-obese) individuals will become obese again However it isalso plausible to consider that 0 lt 120590 lt 1 which representsthat ex-obese population is more resistant to being obeseagain that is they are more aware of the health risks thatthis would represent The consideration that 0 lt 120590 lt 1 doesnot change the results of the stability analysis and periodicorbits performed in Sections 3 and 4 With respect to thelatter the nullcline 1198661 (given by (33)) changes to 1198661015840
1 where1198661015840
1= minus1198661 while 1198662 (34) remains the same The intersection
of the nullclines 11986610158401and 1198662 again occurs at only one point
in the region Δ and its stability is the same Regarding thisaspect we could conclude that regardless of the value ofthe parameter 120590 there is always the possibility that the ex-obese population is at risk of relapse and becoming obeseagain of course this possibility is less if 0 lt 120590 lt 1 andgreater if 120590 gt 1 In this sense public health programs againstobesity designed by governments may be more effectiveif they take into account the awareness of the ex-obesepopulation
Finally with regard to the mentioned relapse term120590[120573119868(119905) + 120598]119877(119905) if we consider only the effects of socialcontagion (which corresponds to 120598 = 0) it is reduced tothe nonlinear term 120590120573119868(119905)119877(119905) which causes system (4) toexhibit a backward bifurcation If the effects of nonsocialcontagion (that is 120598 gt 0) are also incorporated the presenceof the linear term 120590120573120598119877(119905) causes the backward bifurcationto disappear in system (4) and in its place it only presentsan equilibrium point global asymptotically stable The lattercase suggests that the way in which the effects of nonsocialcontagion are introduced into the model by means of thelinear part of the relapse term should be modified in orderto maintain much of the interesting dynamic that is presentin the backward bifurcation Certainly an alternative wayis considering instead of a linear a nonlinear contribution
12 Discrete Dynamics in Nature and Society
Perhaps it is possible to build amore general nonlinearmodelwhere in the case in which the contribution due to nonsocialeffects gradually fades the backward bifurcation of the Ejimaet al model could be obtained as a limit case
Competing Interests
The authors declare that they have no conflict of interests
Acknowledgments
This research was supported by the UACM-SECITI ResearchGrant PI-2013-34 Cruz Vargas-De-Leon would like toexpress his indebtedness to Dr Ramon Reyes Carreto for hishospitality during his stay as visiting professor at the UAGroin Chilpancingo
References
[1] World Health Organization Obesity Preventing and Managingthe Global Epidemic Report of a WHO Consultation WHOTechnical Report Series No 894 WHO Geneva Switzerland2000 httpswwwgooglecommxurlsa=tamprct=jampq=ampesrc=sampsource=webampcd=1ampved=0ahUKEwjlno 3tsDRAhVqi1QKH-W0rBPIQFggZMAAampurl=http3A2F2Fwhqlibdocwhoint2Ftrs2FWHO TRS 894pdfampusg=AFQjCNFLusNXrJz-xFKX1xpQwkHLEr-Lohgampbvm=bv144224172damcampcad=rja
[2] C L Vanvrancken-Tompkins andMaM Sothern ldquoPrevencionde la obesidad en los ninos desde el nacimiento hasta loscincoanosrdquo in Enciclopedia Sobre el Desarrollo de la PrimeraInfancia R Tremblay R Barr R Peters and M Boivin Edsvol 1ndash7 Centre of Excellence for EarlyChildhoodDevelopmentQuebec Canada 2010
[3] S K Garg H Maurer K Reed and R Selagamsetty ldquoDiabetesand cancer two diseases with obesity as a common risk factorrdquoDiabetes Obesity andMetabolism vol 16 no 2 pp 97ndash110 2014
[4] National Heart ldquoLung and Blood Institute World Health WhatAre the Health Risks of Overweight and Obesityrdquo httpswwwnhlbinihgovhealthhealth-topicstopicsoberisks
[5] World Health Organization (WHO) Global Health Observa-tory Data Repository WHO Geneva Switzerland 2011 httpappswhointghodata
[6] World Health Organization The Global Strategy on Diet Phys-ical Activity and Health (DPAS) WHO Geneva Switzerland2004 httpwwwwhointnmhwha59dpasen
[7] N A Christakis and J H Fowler ldquoThe spread of obesity in alarge social network over 32 yearsrdquoThe New England Journal ofMedicine vol 357 no 4 pp 370ndash379 2007
[8] K Ejima K Aihara and H Nishiura ldquoModeling the obesityepidemic social contagion and its implications for controlrdquoTheoretical Biology and Medical Modelling vol 10 article 17 pp1ndash13 2013
[9] A J Arenas G Gonzalez-Parra and L Jodar ldquoPeriodic solu-tions of nonautonomous differential systems modeling obesitypopulationrdquo Chaos Solitons amp Fractals vol 42 no 2 pp 1234ndash1244 2009
[10] G Gonzalez-Parra L Acedo R-J Villanueva Mico and A JArenas ldquoModeling the social obesity epidemic with stochasticnetworksrdquo Physica A Statistical Mechanics and its Applicationsvol 389 no 17 pp 3692ndash3701 2010
[11] G Gonzalez-Parra L Jodar F J Santonja and R J VillanuevaldquoAn age-structured model for childhood obesity mathematicalpopulation studiesrdquo An International Journal of MathematicalDemography vol 17 no 1 pp 1ndash11 2010
[12] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 Article ID e1000968 2010
[13] L Jodar F J Santonja and G Gonzalez-Parra ldquoModelingdynamics of infant obesity in the region of Valencia SpainrdquoComputers andMathematics with Applications vol 56 no 3 pp679ndash689 2008
[14] F-J Santonja R-J Villanueva L Jodar and G Gonzalez-Parra ldquoMathematical modelling of social obesity epidemic inthe region of Valencia Spainrdquo Mathematical and ComputerModelling of Dynamical Systems vol 16 no 1 pp 23ndash34 2010
[15] F J Santonja and L Shaikhet ldquoProbabilistic stability analysisof social obesity epidemic by a delayed stochastic modelrdquoNonlinear Analysis Real World Applications vol 17 pp 114ndash1252014
[16] D Tudor ldquoA deterministic model for herpes infections inhuman and animal populationsrdquo SIAM Review vol 32 no 1pp 136ndash139 1990
[17] A M Lyapunov The general problem of the stability of motionTaylor amp Francis London England 1992
[18] C Vargas-De-Leon ldquoLyapunov functions for two-species coop-erative systemsrdquo Applied Mathematics and Computation vol219 no 5 pp 2493ndash2497 2012
[19] D W Jordan and P Smith Nonlinear Ordinary DifferentialEquations An Introduction for Scientists and Engineers OxfordUniversity Press New York NY USA 4th edition 2007
[20] C M Kribs-Zaleta and J X Velasco-Hernandez ldquoA simplevaccination model with multiple endemic statesrdquoMathematicalBiosciences vol 164 no 2 pp 183ndash201 2000
[21] S Barquera I Campos-Nonato L Hernandez-Barrera APedroza-Tobıas and J A Rivera-Dommarco ldquoPrevalence ofobesity in mexican adults ensanut 2012rdquo Salud Publica deMexico vol 55 supplement 2 pp S151ndashS160 2013
whose eigenvalues are 1205821 = minus120583 and 1205822 = minus120583(120574 + 120583)(1 minus 1198770)Note that if 1198770 lt 1 then 1205822 lt 0 and therefore 1198640 is locallyasymptotically stable while when 1198770 gt 1 1205822 gt 0 and 1198640 isunstable In both cases the equilibrium point is hyperbolicThis situation is depicted in Figure 2 by the horizontal line onthe 1198770 axis wherein for 1198770 lt 1we have stability (indicated bya solid line) and for 1198770 gt 1 instability (showed by a dashedline) It should be noted that if 1198770 = 1 then 1205822 = 0 asa consequence the equilibrium point is nonhyperbolic and1198770 = 1 could be the value in which a bifurcation is producedHence we have the following result
Theorem 1 If 1198770 lt 1 then 1198640 is an equilibrium of system (6)and it is locally asymptotically stable Otherwise if1198770 gt 1 then1198640 is unstable(2)The Endemic Equilibria 119864lowast = (119877lowast 119868lowast) By substituting theendemic equilibria 119864lowast = (119868lowast 119877lowast) in system (6) the identitiesare obtained (120590 minus 1) 120573119877lowast = 120573119868lowast minus 120573119873 + (120574 + 120583)
120583119877lowast = 120574119868lowast minus 120590120573119868lowast119877lowast (14)
which allows writing the Jacobian matrix (12) evaluated insuch points as
119869 (119868lowast 119877lowast) = (minus120573119868lowast (120573119868lowast minus (120574 + 120583) (1 minus 1198770)) 119868lowast119877lowast120583119877lowast119868lowast minus120590120573119868lowast minus 120583 ) (15)
Det (119869) equiv 119886119868lowast2 minus 119888 (18)
are the trace and determinant of (15) respectively Becausein the Δ region both coordinates of 119864lowast are positive thenTr(119869) always is negative not so with Det(119869) which could bepositive or negative According to the discriminant 120575 of (11)in the corresponding analysis of the equilibrium points inorder to ensure that 119864lowast isin Δ the following three cases maybe identified
(i) If 120575 lt 0 then considering in this condition expres-sions 119886 119887 and 119888 given by (8) we obtain 1198770 lt 119877lowast
0
where
119877lowast0equiv (120590 minus 1) 120583 + 120574 + 2radic(120590 minus 1) 120583120574120590 (120574 + 120583) (19)
Consequently in accordance with (11) in this intervalwe have that there is not endemic equilibrium pointsSince 120583 and (120590minus1)120574 are two positive real numbers the
relation between its arithmetic and geometric meansis always given by
12 [(120590 minus 1) 120574 + 120583] ge radic(120590 minus 1) 120574120583 (20)
therefore from (19) and (20) itmay be concluded that119877lowast0le 1
(ii) If 120575 = 0 then substituting 119886 119887 and 119888 given by (8)in this equality we obtain 1198770 = 119877lowast
0 Thus according
to (11) when 1198770 reaches this value we have only oneendemic equilibrium point and 119868lowast = 119868lowast
0 where
119868lowast0equiv radic120574120583 (120590 minus 1) minus 120583120590120573 (21)
is the corresponding infected population of119877lowast0More-
over if (19) and (21) are substituted in determinant(18) and trace (17) these are simplified as
Det (119869) = 0 (22)
Tr (119869) = minus(120590 + 1)radic120574120583 (120590 minus 1) minus 120583120590 (23)
respectively Consequently taking into account (22)and (23) eigenvalues (16) are reduced to 1205821 = Tr(119869) lt0 and 1205822 = 0 one of which is negative and theother is zero Therefore this equilibrium point isnonhyperbolic and 1198770 = 119877lowast
0could also be the value
in which a bifurcation is produced See Figure 2(iii) If 120575 gt 0 then similarly using 119886 119887 and 119888 given by (8)
this inequality leads to 119877lowast0lt 1198770 As a consequence
from (11) it follows that in this interval we havefamilies of endemic equilibrium points determinedby two quadratic branches It may be shown that forall endemic equilibrium points that form the uppercurve since 119868lowast gt 119868lowast
0 from (18) it results in the fact
that Det(119869) gt 0 thereby in (16) it follows that botheigenvalues 1205821 and 1205822 are negative and thereforethese equilibriumpoints are asymptotically stable Onthe other hand it can also be shown that for endemicequilibrium points that are part of the lower curvesince 119868lowast lt 119868lowast
0 from (18) it occurs that Det(119869) lt 0
in this way again from (16) it is obtained that 1205821 gt 0and1205822 lt 0 and consequently such equilibriumpointsare unstable Note that in the latter case (in orderthat 119864lowast isin Δ) quadratic branch is delimited by thepoint (1 0) in which by the way Det(119869) = 0 andtherefore 1205821 lt 0 and 1205822 = 0 that is the said point isnonhyperbolic and in it a bifurcation could also occurThese results are illustrated in Figure 2
It is necessary to point out that from the above arguments itfollows that if 0 lt 1198770 lt 119877lowast
0 then the disease-free family of
equilibrium points 1198640 is the only one in the feasible regionΔ Based on the previous analysis we have the followingtheorem which summarizes the local stability of all endemicequilibria points of system (6) contained in Δ
Discrete Dynamics in Nature and Society 5
Theorem 2 If 0 lt 1198770 lt 119877lowast0 then there are not endemic
equilibrium points 119864lowast in the feasible region Δ Also If 1198770 =119877lowast0 then in such region there is a unique 119864lowast which is not
hyperbolic Moreover if 119877lowast0
lt 1198770 then there are familiesof endemic equilibrium points determined by two quadraticbranches an upper in which all points are asymptoticallystable and the other lower formed by unstable points and withepidemiological significance only when 1198770 ⩽ 1
In Figure 2 the results indicated by Theorems 1 and 2are displayed schematically There it may be appreciated thatthe trajectories of solutions of system (4) change abruptly inthe nonhyperbolic points (119877lowast
0 119868lowast) and (1 0) that is in these
points really bifurcations occur In that sense Figure 2 can beconsidered as the bifurcation diagram in the feasible regionΔ of said system and the type of bifurcation shown in itcorresponds to a backward bifurcation
32 Nonexistence of Periodic Orbits and Global StabilityGlobal analysis of system (6) includes the study of periodicorbits The following result shows that this system does nothave periodic orbits
Theorem 3 System (6) does not have periodic orbits in theinterior of ΔProof Consider system (6) for 119868 gt 0 and 119877 gt 0 We considerthe Dulac function
that is 120597(Φ119875)120597119868 + 120597(Φ119876)120597119877 lt 0 in the interior of Δit follows from the Dulac criterion that system (6) has noperiodic orbits in that region
On the other hand global analysis of system (6) alsoinvolves the study of global stability of its disease-freeequilibrium and its endemic equilibria The former may beproved by arguments of local stability as can be seen in thefollowing theorem
Theorem 4 If 0 lt 1198770 lt 119877lowast0 then the unique disease-free
equilibrium 1198640 of system (6) is globally asymptotically stablein ΔProof According to the discussion given previously duringthe formulation of Theorems 1 and 2 it was found that if0 lt 1198770 lt 119877lowast
0 then the disease-free equilibrium point 1198640
is the only one and it also is asymptotically stable in thefeasible region Δ Besides according to Theorem 3 there arenot periodic orbits here As a result all initial condition inΔ which satisfies this interval originates paths tending to1198640 consequently such point must be globally asymptoticallystable
Finally in order to prove the global stability of a uniqueendemic equilibrium 119864lowast in the interior of Δ when 1198770 gt 1 weuse the method of Lyapunov functions This is accomplishedin the next theorem
Theorem5 Assume that1198770 gt 1Then a unique endemic equi-librium 119864lowast = (119868lowast 119877lowast) of system (6) is globally asymptoticallystable in the interior of ΔProof To prove the global asymptotic stability of the uniqueendemic equilibrium 119864lowast we define a Lyapunov function119882(119868 119877) isin R2
0+ 119868 gt 0 119877 gt 0 rarr R given by
119882(119868 119877) = 120583(ln 119868119868lowast + 119868lowast119868 minus 1)+ 120573 (120590 minus 1) (119877 minus 119877lowast minus 119877lowast ln 119877119877lowast )
(26)
Let119882119894 = ln(119868119868lowast) + 119868lowast119868 minus 1 By using 120573119873 minus (120574 + 120583) = 120573119868lowast minus120573(120590 minus 1)119877lowast we have119889119882119894119889119905 = 120573119868lowast (2 minus 119868lowast119868 minus 119868119868lowast)
+ 120573 (120590 minus 1) 119877lowast ( 119877119877lowast minus 1 minus 119868lowast119877119868119877lowast + 119868lowast119868 ) (27)
Let119882119903 = 119877minus119877lowastminus119877lowast ln(119877119877lowast) By using 120574 = 120590120573119877lowast+120583(119877lowast119868lowast)we have
119889119882119903119889119905 = 120590120573119877lowast119868 (2 minus 119877119877lowast minus 119877lowast119877 )+ 120583119877lowast ( 119868119868lowast minus 119877119877lowast minus 119868119877lowast119868lowast119877 + 1) (28)
The derivative of (26) along solution of (6) is given by
119889119882119889119905 = 120583119889119882119894119889119905 + 120573 (120590 minus 1) 119889119882119903119889119905 (29)
By means of (27) and (28) we obtain
119889119882119889119905 = 120583120573 (119868lowast minus (120590 minus 1) 119877lowast) (2 minus 119868lowast119868 minus 119868119868lowast)+ 120573 (120590 minus 1) 120590120573119877lowast119868 (2 minus 119877119877lowast minus 119877lowast119877 )+ 120583120573 (120590 minus 1) 119877lowast (2 minus 119868119877lowast119868lowast119877 minus 119868lowast119877119868119877lowast ) lt 0
(30)
Using
120573119868lowast minus 120573 (120590 minus 1) 119877lowast = 120573119873 minus (120574 + 120583)= (120574 + 120583) (1198770 minus 1) (31)
6 Discrete Dynamics in Nature and Society
we have119889119882119889119905 = 120583 (120574 + 120583) (1198770 minus 1) (2 minus 119868lowast119868 minus 119868119868lowast)+ 120573 (120590 minus 1) 120590120573119877lowast119868 (2 minus 119877119877lowast minus 119877lowast119877 )+ 120583120573 (120590 minus 1) 119877lowast (2 minus 119868119877lowast119868lowast119877 minus 119868lowast119877119868119877lowast ) lt 0
(32)
Therefore if 1198770 gt 1 then 119889119882119889119905 is negative definite Bythe Lyapunov asymptotic stability theorem [17] this resultimplies that119864lowast is globally asymptotically stable in the interiorof ΔRemark 6 Recently Vargas-De-Leon [18] used the Lyapunovfunction119882119894 = ln(119868119868lowast)+119868lowast119868minus1 to prove global stability of thecoexistence equilibrium of two-species mutualism models
4 Case with Social and Nonsocial ContagionRisks of Obesity (120598 gt 0)
41 Equilibria and Nonexistence of Periodic Orbits Wemightfollow the usual method of setting the right-hand side of (4)equal to zero in order to obtain their equilibria This proce-dure is not the most appropriate because it leads to a cubicequation whose analytical solutions are quite complicatedInstead an alternative way to find the equilibrium points isto perform a geometric analysis based on the intersection ofthe nullclines of system (4) The nullclines = 0 (verticaldirections) and = 0 (horizontal directions) are given bythe functions
1198661 (119868) = 1205731198682 minus (120573119873 minus 120598 minus 120583 minus 120574) 119868 minus 120598119873120573 (120590 minus 1) 119868 + 120598 (120590 minus 1) (33)
respectivelyNote that1198661(119868) and1198662(119868) respectively are discontinuous
in 119868 = minus120598120573 and 119868 = minus(120590120598 + 120583)120590120573 Besides it is clear that1198661(119868) rarr +infin (resp 1198662(119868) rarr 120574120590120573) as 119868 rarr +infin and that thefunction1198661(119868) (resp1198662(119868)) is increasing and concave downGiven that 1198661(0) = minus119873(120590 minus 1) lt 0 and 1198662(0) = 0 thenthe functions 1198661(119868) and 1198662(119868) intersect at a single point (theequilibrium point) in the first quadrant (see Figure 3) Thisresult indicates that in Δ an endemic equilibrium point existsand is unique
Regarding whether or not system (4) has periodic orbitswe have precisely the following result
Theorem 7 System (4) does not have periodic orbits in theinterior of ΔProof Consider system (4) for 119868 gt 0 and 119877 gt 0 From theDulac function
Φ (119868 119877) = 1119868 (35)
Equilibrium point
minus120590120576 + 120583
120590120573minus120576
120573
minusN
120590 minus 1
120574
120590120573
R
G1(I)
G2(I)
I
Figure 3 Schematic representation of a branch of the verticalnullcline 1198661(119868) (dashed line) and the horizontal nullcline 1198662(119868)(solid line) Both curves intersect inΔ once at the equilibrium point
Equilibriumpoint
R
(0 N)
R = N minus I
I+ (N 0) I
Figure 4 Trapping region bounded by the dashed line formed bythe horizontal and vertical axes and the diagonal line 119877 = 119873 minus 119868This triangular region contains inside the only equilibrium point
it follows that120597 (Φ119875 (119868 119877))120597119868 + 120597 (Φ119876 (119868 119877))120597119877= minus 1205981198682 (119873 minus (119868 + 119877)) minus (120573 + 120598119868) minus 1205901205981198771198682minus 120590(120573 + 120598119868) minus 120583119868
(36)
Since 120597(Φ119875)120597119868+120597(Φ119876)120597119877 lt 0 inΔ taking into account theDulac criterion we conclude that system (4) has no periodicorbits
42 Local andGlobal Stability In order to determine the localand global stability of the single equilibrium point in Δ weconsider that it is inside of the triangular region bounded bythe dashed line shown in Figure 4 This is a trapping region
Discrete Dynamics in Nature and Society 7
the vector field of system (4) on the boundary point into thebox For the points on the horizontal side = 120574119868 gt 0 and = minus(119868 minus 119868+)(119868 minus 119868minus) where 119868+ and 119868minus are the positive andnegative roots of (120573119868 + 120598)(119868 minus 119873) + (120574 + 120583)119868 respectivelyThus these inequalities imply that the vector field points upand right on the interval (0 119868+) while it goes up and left on(119868+ 119873) Moreover in the vertical side = minus(120590120598 + 120583)119877 le 0and = 120598119873 + 120598(120590 minus 1)119877 gt 0 This implies that the vector fieldis directed downward and right except at the origin where itgoes to the right
Finally on the diagonal side of slope minus1 extending fromthe point (0119873) to (119873 0) it can be shown that
119889119877119889119868 = minus1 + 120583119873(119868 minus 119868lowast+) (119868 minus 119868lowast
minus) (37)
where 119868lowast+and 119868lowastminusare the positive and negative roots of 120590(120573119868 +120598)(119868 minus 119873) + (120574 + 120583)119868 respectively Hence the vector field
for large values of 119868 is almost parallel to the diagonal lineIn a more precise analysis always 119868 minus 119868lowast
minusgt 0 and 119868 minus 119868lowast
+gt0 if 119868 gt 119868lowast
+or 119868 minus 119868lowast
+lt 0 if 119868 lt 119868lowast
+ According to
(37) the first case implies that the vector field points inward(to the right) on the diagonal line because 119889119877119889119868 is morenegative than minus1 while the latter indicates that the vectorfield is directed inside (to the left) the diagonal side because119889119877119889119868 is less negative than minus1 (See Figure 4) Therefore weconclude that the triangular region is effectively a trappingregion
On the other hand Theorem 7 prohibits the existenceof closed orbits inside the triangular region Consequentlythe trajectories entering the triangular region accordingto the Poincare-Bendixson theorem [19] must converge tothe single equilibrium point located inside it Thereforethe equilibrium point inside the triangular region must beasymptotically stable Moreover since this point is the onlypoint of equilibrium in Δ it is also globally asymptoticallystable
The results of this section may be summarized in thefollowing theorem
Theorem 8 System (4) has a unique equilibrium point in thetriangular region Δ which is globally asymptotically stable
5 Numerical Results
In this section we perform a series of numerical simulationsand graphs of system (4) to illustrate the different resultsobtained for each of the two cases of interest 120598 gt 0 and120598 = 0 previously analyzed We will use the parameter valuesreported in the work developed by Ejima et al [8] theirepidemiological meanings and magnitudes are indicated inTable 1
51 Case 120598 = 0 According to the magnitudes of theparameters reported in Table 1 (19) takes the value 119877lowast
0=069363 whereas (21) acquires the value 119868lowast
0= 163304
Table 1 Baseline values and epidemiological meanings of theparameters given by Ejima et al [8]
Parameter Baseline values Epidemiological meaning
119873 100000 individuals(assumed) Population size
1120583 694 (per year) Average life expectancy at birth120573 296 times 10minus7 (peryear)
Transmission rate of obesity dueto social contagion120598 0012 (per year) Nonsocial contagion risk ofobesity120590 80 Relative hazard of obesity amongthe ex-obese1120574 358 (per year) Average duration of obesity
2000
20000
0
4000
4000
6000
600080
00
8000
1000
0
10000
1200
0
12000
1400
0
14000
1600
0
16000
1800
0
18000
2000
0
200002200024000260002800030000
2200
024
000
2600
028
000
3000
0
I (obese)
R (e
x-ob
ese)
Figure 5 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 03 obtained by considering 120573 = 127 times 10minus7Furthermore taking into consideration (10) and (11) theresulting two endemic equilibria are119864lowast
1= (20918 91378) (38)
119864lowast2= (12522 79384) (39)
which as discussed previously are asymptotically stable andunstable respectively
In Figures 5ndash9 we illustrated representative phase por-traits of the dynamic system (6) for some different values of1198770 between 0 and just over 1 according to the bifurcationdiagram shown in Figure 2 These values are obtained bymaintaining fixed in (9) 119873 120583 and 120574 but varying theparameter 120573 Thus if 1198770 takes values between 0 and 119877lowast
0
say 1198770 = 03 we have only one equilibrium point (thedisease-free equilibrium) which is globally asymptoticallystable (see Figure 5) If 1198770 = 119877lowast
0 we have two equilibrium
points one asymptotically stable disease-free equilibriumand the other endemic equilibria point which is nonhyper-bolic (see Figure 6) When 1198770 takes values between 119877lowast0 and
8 Discrete Dynamics in Nature and Society
2000
2000
0
0
4000
4000
6000
6000
8000
800010
000
10000
1200
0
12000
1400
0
14000
1600
0
16000
1800
0
18000
2000
0
20000
2200
0
2400
0I (obese)
R (e
x-ob
ese)
Figure 6 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 119877lowast0 obtained by considering 120573 = 294 times 10minus7
5000
2000
0
0
1000
0
4000
1500
0
6000
2000
0
8000
2500
0
10000
3000
0
12000
3500
0
14000
4000
0
16000
4500
0
18000
5000
0
20000
I (obese)
R (e
x-ob
ese)
Figure 7 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 08 obtained by considering 120573 = 339 times 10minus71 say 1198770 = 08 we have three equilibria one disease-free equilibrium (asymptotically stable) and two endemicequilibria (asymptotically stable and unstable) as shown inFigure 7 If 1198770 = 1 there are two equilibrium pointsone disease-free equilibrium which is nonhyperbolic and theother asymptotically stable endemic equilibrium as shown inFigure 8 Finally if 1198770 takes a slightly greater value to 1 forexample1198770 = 12 we have again two equilibrium points onedisease-free equilibrium (unstable) and the other endemicequilibrium (asymptotically stable) but both are hyperbolic(see Figure 9)
On the other hand in Figures 10ndash14 examples of theevolution in time of 119868(119905) and119877(119905) are presented for the values1198770 = 03 1198770 = 119877lowast
0 1198770 = 08 1198770 = 1 and 1198770 = 12 which
1000
00
2000
0
3000
0
4000
0
5000
0
6000
0
7000
0
8000
0
9000
0
1000
00
1100
00
1200
00
I (obese)
10000
20003000400050006000700080009000
10000110001200013000140001500016000
R (e
x-ob
ese)
Figure 8 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 1 obtained by considering 120573 = 423 times 10minus7
1000
0
10000
0
2000
0
2000
3000
0
30004000
4000
050
000
50006000
6000
07000
7000
0
80009000
8000
090
000
1000011000
1000
00
12000
1100
00
1400013000
1200
0013
0000
1400
00
1500016000
I (obese)
R (e
x-ob
ese)
Figure 9 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 12 obtained by considering 120573 = 508 times 10minus7illustrate respectively some of the dynamics shown in thephase portraits in Figures 5ndash9
52 Case 120598 gt 0 Taking into account the value 120598 =0012 gt 0 given in Table 1 in addition to the othersthere reported nullclines (33) and (34) intersect at the onlyendemic equilibrium point119864lowast = (60890 66805) (40)
which is globally asymptotically stable In Figure 15 thecorresponding phase portrait with several representativetrajectories around this equilibrium point is shown Also
Discrete Dynamics in Nature and Society 9
Time (years)
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 10 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 03 and the initial condition is(119868(0) 119877(0)) = (10000 30000)
Time (years)0 100 200 300
ObeseEx-obese
20000
15000
10000
5000
0
Figure 11 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 119877lowast
0and the initial condition is(119868(0) 119877(0)) = (20000 8000)
performing another series of numerical simulations of system(4) we show for this case in Figure 16 an example of theevolution in time of 119868(119905) and 119877(119905)6 Discussion and Conclusions
According to the results of our analysis of the Ejima et almodel we arrive at the following conclusions It has beenshown that the Ejima et al model [8] if only the socialcontagion (120598 = 0) is considered presents the two bifurcationvalues 1198770 = 119877lowast
0and 1198770 = 1 which results in what is
known as backward bifurcation From a mathematical pointof view the backward bifurcation is apparently constitutedby the combination of two different kinds of bifurcations
Time (years)
40000
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 12 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 08 and the initial condition is(119868(0) 119877(0)) = (30000 12000)
Time (years)
40000
50000
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 13 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 1 and the initial condition is(119868(0) 119877(0)) = (18000 2500)a saddle-node bifurcation that occurs at 1198770 = 119877lowast
0 whose
upper (asymptotically stable) and lower (unstable) quadraticbranches are defined for 119877lowast
0lt 1198770 and a transcritical
bifurcation that takes place in 1198770 = 1 whose horizontalbranches on the 1198770 axis are asymptotically stable for 1198770 lt 1and unstable for 1 lt 1198770 For values of 1 lt 1198770 only theupper quadratic branch asymptotically stable of the saddle-node bifurcation and the unstable horizontal branch of thetranscritical bifurcation have ameaning in the feasible regionΔ The diagram shown in Figure 2 suggests the presenceof these two bifurcations in the vicinities of the points(119877lowast0 119868lowast0) and (1 0) Of course we should go beyond the
graphic evidence and employ the qualitative analysis of localbifurcations in those points to verify our statement even
10 Discrete Dynamics in Nature and Society
Time (years)0 100 200 300
ObeseEx-obese
140000
120000
100000
80000
60000
40000
20000
0
Figure 14 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 12 and the initial condition is(119868(0) 119877(0)) = (100000 80000)
1000
0
10000
0
2000
0
2000
3000
0
30004000
4000
0
5000
0
50006000
6000
0
7000
7000
0
80009000
8000
0
9000
0
1000011000
1000
00
12000
1100
00
1400013000
1200
00
1300
00
1400
00
1500016000
I (obese)
R (e
x-ob
ese)
Figure 15 Phase portrait for system (4) for the case 120598 = 0012 gt 0(considering 120573 = 296 times 10minus7)without the realization of this depth analysis our discussionsand conclusions can be supported firmly from the resultsshown in this work of investigation
Indeed the presence of the different branches of familiesof equilibrium points shown in Figure 2 causes the system todisplay as the parameter1198770 is varied a bistable region and asa consequence the hysteresis phenomenon In order to showin more detail the presence of hysteresis in the bifurcationdiagram mentioned before suppose we start the system inthe state119860 (see Figure 17) and then slowly increase 1198770 (whichis indicated by an arrow to the right just beneath the axis1198770 of Figure 17) We remain at the origin until 1198770 = 1when the origin loses its stability At this point at the slightestdisturbance the system will jump to the state 119861 located in the
0
20000
40000
60000
80000
100000
120000
Time (years)0 100 200 300
ObeseEx-obese
Figure 16 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0012 gt 0 and the initial condition is(100000 50000)
B
A
1
C
80000
70000
60000
50000
40000
30000
20000
10000
0
Infe
cted
(I)
Rlowast0
Basic reproductive number (R0)
Figure 17 Schematic representation of hysteresis phenomenon thatexhibits the bifurcation diagram of system (4) for the case 120598 = 0Here the forward and backward paths are not equal as the parameter1198770 is changedstable upper branch Insofar as1198770 increases its value (1198770 gt 1)the equilibrium points of the system will move along thisbranch away from the state119861 If now1198770 continuously reducedits value equilibrium states will return to the state 119861 cross itand reach the state 119862 (in which 1198770 = 119877lowast
0) At this point at
the slightest change again equilibrium states will jump backto the origin (in 1198770 = 119877lowast0 ) and if 1198770 continues to decrease wewill return to the state119860 Consequently the system exhibits alack of reversibility or it is said that it has memory (or that itpresents hysteresis) because the forward and backward pathsare not identical
From an epidemiological point of view in this backwardbifurcation if 0 lt 1198770 lt 119877lowast
0 no matter where the initial
Discrete Dynamics in Nature and Society 11
conditions are taken solutions always tend to the straightline of stable disease-free points (that correspond to a never-obese population) When 119877lowast
0lt 1198770 lt 1 if the initial
conditions are located below the unstable quadratic branchof endemic equilibrium points (obese people) then solutionscontinue going to the horizontal line of stable disease-freepoints otherwise they will be directed to the stable curveof endemic points located at the upper quadratic branchFinally if 1 lt 1198770 no matter again where the initialconditions are situated solutions will continue heading forthe stable endemic points of such upper branch Should benoted that unlike what happens in systems that exhibit thephenomenon of forward bifurcation in which the endemicequilibrium exists only for 1198770 gt 1 [20] in systems exhibitinga backward bifurcation under certain initial conditions itis possible to have endemic states when 1198770 lt 1 In ourcase the endemic equilibrium that exists for 1198770 just aboveone has a large obese population so the result of 1198770 risingabove one would be a sudden and dramatic jump in thenumber of infective Moreover reducing 1198770 back below onewould not eradicate the obesity if the size of the infectedpopulation is greater or equal than 119868lowast
0 the corresponding
value of 119877lowast0 In order to eradicate the disease 1198770 must be
further reduced so it just needs to be slightly less than119877lowast0 to suddenly enter the region where only stable disease-
free equilibrium (never-obese population) exists Both suchdifferent abrupt behaviours are the result of hysteresisphenomenon
It can be seen from the bifurcation diagram shown inFigure 2 that theway inwhich obesity infected population canbe lowered (or raised) is through the reduction (or increase)of 1198770 parameter which in accordance with (9) depends onthe four quantities 119873 120573 120583 and 120574 Since we have assumedin (1) that the natural death and birth rates are equal anddenoted by 120583 the population size119873 is constantTherefore1198770can only change according to the following forms Firstly 1198770diminishes if the natural rate 120583 (of death or birth) increasesthe transmission rate 120573 decreases or the rate 120574 grows atwhich the infected individuals become recovered On theother hand 1198770 increment its value if the opposite occurs 120583decreases 120573 goes up or 120574 declines It should be noted thatchanges in 120583 could be slower and achieved in the long termprobably in decades which is not a viable option Converselychanges in 120573 or 120574 might be faster and perhaps occur infewer years Since the focus is on reducing obesity levels andtherefore the values of 1198770 it is of interest to seek mechanismsto decrease120573 andor increase 120574The former could be achievedby sensitizing at the population in general about the benefitsassociatedwith having a healthy and balanced diet and also bydesigning effective health and informative public preventionpoliciesThe latter could be done by providing better medicalassistance and making it available to the obese populationIt should be emphasized that this last strategy is not feasiblein countries with high levels of obesity as Mexico because atthe present time it is very expensive and will become evenmore expensive as the affected population increases In thisregard we believe that it is much more convenient that thegovernments of the affected countries implement broaderand more effective preventive policies
It is worth mentioning that in Mexico there have beenstudies to quantify more precisely the prevalence of peoplesuffering from overweight and obesity It is estimated that713 of Mexican adults aged 20 or more are found inthis condition [21] While trends show slowing down of theincrease in the obesity prevalence there is no evidence toinfer that prevalence will decrease in the coming years Inthis sense studies like this allow us to analyze the trends ofthe population with obesity and study the different possiblescenarios Particularly with the model studied this couldbe done for Mexico if we knew accurately the values ofits parameters 120573 and 120574 To our knowledge there is notsuch information Nevertheless these parameters could beestimated in a short or medium term based on statisti-cal information collected on obesity in our country Theknowledge of how these parameters behave over time wouldcontribute in quantifying the effectiveness of the Mexicanpublic policies against obesity
With respect to the model studied it is pertinent tocomment about the term of relapse 120590[120573119868(119905) + 120598]119877(119905) of thesecond equation (1) In the first place in this work it has beenconsidered that the parameter 120590 the relative risk of weightregain among ex-obese individuals is greater than unity (120590 gt1)This condition represents the high risk that recovered (ex-obese) individuals will become obese again However it isalso plausible to consider that 0 lt 120590 lt 1 which representsthat ex-obese population is more resistant to being obeseagain that is they are more aware of the health risks thatthis would represent The consideration that 0 lt 120590 lt 1 doesnot change the results of the stability analysis and periodicorbits performed in Sections 3 and 4 With respect to thelatter the nullcline 1198661 (given by (33)) changes to 1198661015840
1 where1198661015840
1= minus1198661 while 1198662 (34) remains the same The intersection
of the nullclines 11986610158401and 1198662 again occurs at only one point
in the region Δ and its stability is the same Regarding thisaspect we could conclude that regardless of the value ofthe parameter 120590 there is always the possibility that the ex-obese population is at risk of relapse and becoming obeseagain of course this possibility is less if 0 lt 120590 lt 1 andgreater if 120590 gt 1 In this sense public health programs againstobesity designed by governments may be more effectiveif they take into account the awareness of the ex-obesepopulation
Finally with regard to the mentioned relapse term120590[120573119868(119905) + 120598]119877(119905) if we consider only the effects of socialcontagion (which corresponds to 120598 = 0) it is reduced tothe nonlinear term 120590120573119868(119905)119877(119905) which causes system (4) toexhibit a backward bifurcation If the effects of nonsocialcontagion (that is 120598 gt 0) are also incorporated the presenceof the linear term 120590120573120598119877(119905) causes the backward bifurcationto disappear in system (4) and in its place it only presentsan equilibrium point global asymptotically stable The lattercase suggests that the way in which the effects of nonsocialcontagion are introduced into the model by means of thelinear part of the relapse term should be modified in orderto maintain much of the interesting dynamic that is presentin the backward bifurcation Certainly an alternative wayis considering instead of a linear a nonlinear contribution
12 Discrete Dynamics in Nature and Society
Perhaps it is possible to build amore general nonlinearmodelwhere in the case in which the contribution due to nonsocialeffects gradually fades the backward bifurcation of the Ejimaet al model could be obtained as a limit case
Competing Interests
The authors declare that they have no conflict of interests
Acknowledgments
This research was supported by the UACM-SECITI ResearchGrant PI-2013-34 Cruz Vargas-De-Leon would like toexpress his indebtedness to Dr Ramon Reyes Carreto for hishospitality during his stay as visiting professor at the UAGroin Chilpancingo
References
[1] World Health Organization Obesity Preventing and Managingthe Global Epidemic Report of a WHO Consultation WHOTechnical Report Series No 894 WHO Geneva Switzerland2000 httpswwwgooglecommxurlsa=tamprct=jampq=ampesrc=sampsource=webampcd=1ampved=0ahUKEwjlno 3tsDRAhVqi1QKH-W0rBPIQFggZMAAampurl=http3A2F2Fwhqlibdocwhoint2Ftrs2FWHO TRS 894pdfampusg=AFQjCNFLusNXrJz-xFKX1xpQwkHLEr-Lohgampbvm=bv144224172damcampcad=rja
[2] C L Vanvrancken-Tompkins andMaM Sothern ldquoPrevencionde la obesidad en los ninos desde el nacimiento hasta loscincoanosrdquo in Enciclopedia Sobre el Desarrollo de la PrimeraInfancia R Tremblay R Barr R Peters and M Boivin Edsvol 1ndash7 Centre of Excellence for EarlyChildhoodDevelopmentQuebec Canada 2010
[3] S K Garg H Maurer K Reed and R Selagamsetty ldquoDiabetesand cancer two diseases with obesity as a common risk factorrdquoDiabetes Obesity andMetabolism vol 16 no 2 pp 97ndash110 2014
[4] National Heart ldquoLung and Blood Institute World Health WhatAre the Health Risks of Overweight and Obesityrdquo httpswwwnhlbinihgovhealthhealth-topicstopicsoberisks
[5] World Health Organization (WHO) Global Health Observa-tory Data Repository WHO Geneva Switzerland 2011 httpappswhointghodata
[6] World Health Organization The Global Strategy on Diet Phys-ical Activity and Health (DPAS) WHO Geneva Switzerland2004 httpwwwwhointnmhwha59dpasen
[7] N A Christakis and J H Fowler ldquoThe spread of obesity in alarge social network over 32 yearsrdquoThe New England Journal ofMedicine vol 357 no 4 pp 370ndash379 2007
[8] K Ejima K Aihara and H Nishiura ldquoModeling the obesityepidemic social contagion and its implications for controlrdquoTheoretical Biology and Medical Modelling vol 10 article 17 pp1ndash13 2013
[9] A J Arenas G Gonzalez-Parra and L Jodar ldquoPeriodic solu-tions of nonautonomous differential systems modeling obesitypopulationrdquo Chaos Solitons amp Fractals vol 42 no 2 pp 1234ndash1244 2009
[10] G Gonzalez-Parra L Acedo R-J Villanueva Mico and A JArenas ldquoModeling the social obesity epidemic with stochasticnetworksrdquo Physica A Statistical Mechanics and its Applicationsvol 389 no 17 pp 3692ndash3701 2010
[11] G Gonzalez-Parra L Jodar F J Santonja and R J VillanuevaldquoAn age-structured model for childhood obesity mathematicalpopulation studiesrdquo An International Journal of MathematicalDemography vol 17 no 1 pp 1ndash11 2010
[12] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 Article ID e1000968 2010
[13] L Jodar F J Santonja and G Gonzalez-Parra ldquoModelingdynamics of infant obesity in the region of Valencia SpainrdquoComputers andMathematics with Applications vol 56 no 3 pp679ndash689 2008
[14] F-J Santonja R-J Villanueva L Jodar and G Gonzalez-Parra ldquoMathematical modelling of social obesity epidemic inthe region of Valencia Spainrdquo Mathematical and ComputerModelling of Dynamical Systems vol 16 no 1 pp 23ndash34 2010
[15] F J Santonja and L Shaikhet ldquoProbabilistic stability analysisof social obesity epidemic by a delayed stochastic modelrdquoNonlinear Analysis Real World Applications vol 17 pp 114ndash1252014
[16] D Tudor ldquoA deterministic model for herpes infections inhuman and animal populationsrdquo SIAM Review vol 32 no 1pp 136ndash139 1990
[17] A M Lyapunov The general problem of the stability of motionTaylor amp Francis London England 1992
[18] C Vargas-De-Leon ldquoLyapunov functions for two-species coop-erative systemsrdquo Applied Mathematics and Computation vol219 no 5 pp 2493ndash2497 2012
[19] D W Jordan and P Smith Nonlinear Ordinary DifferentialEquations An Introduction for Scientists and Engineers OxfordUniversity Press New York NY USA 4th edition 2007
[20] C M Kribs-Zaleta and J X Velasco-Hernandez ldquoA simplevaccination model with multiple endemic statesrdquoMathematicalBiosciences vol 164 no 2 pp 183ndash201 2000
[21] S Barquera I Campos-Nonato L Hernandez-Barrera APedroza-Tobıas and J A Rivera-Dommarco ldquoPrevalence ofobesity in mexican adults ensanut 2012rdquo Salud Publica deMexico vol 55 supplement 2 pp S151ndashS160 2013
Theorem 2 If 0 lt 1198770 lt 119877lowast0 then there are not endemic
equilibrium points 119864lowast in the feasible region Δ Also If 1198770 =119877lowast0 then in such region there is a unique 119864lowast which is not
hyperbolic Moreover if 119877lowast0
lt 1198770 then there are familiesof endemic equilibrium points determined by two quadraticbranches an upper in which all points are asymptoticallystable and the other lower formed by unstable points and withepidemiological significance only when 1198770 ⩽ 1
In Figure 2 the results indicated by Theorems 1 and 2are displayed schematically There it may be appreciated thatthe trajectories of solutions of system (4) change abruptly inthe nonhyperbolic points (119877lowast
0 119868lowast) and (1 0) that is in these
points really bifurcations occur In that sense Figure 2 can beconsidered as the bifurcation diagram in the feasible regionΔ of said system and the type of bifurcation shown in itcorresponds to a backward bifurcation
32 Nonexistence of Periodic Orbits and Global StabilityGlobal analysis of system (6) includes the study of periodicorbits The following result shows that this system does nothave periodic orbits
Theorem 3 System (6) does not have periodic orbits in theinterior of ΔProof Consider system (6) for 119868 gt 0 and 119877 gt 0 We considerthe Dulac function
that is 120597(Φ119875)120597119868 + 120597(Φ119876)120597119877 lt 0 in the interior of Δit follows from the Dulac criterion that system (6) has noperiodic orbits in that region
On the other hand global analysis of system (6) alsoinvolves the study of global stability of its disease-freeequilibrium and its endemic equilibria The former may beproved by arguments of local stability as can be seen in thefollowing theorem
Theorem 4 If 0 lt 1198770 lt 119877lowast0 then the unique disease-free
equilibrium 1198640 of system (6) is globally asymptotically stablein ΔProof According to the discussion given previously duringthe formulation of Theorems 1 and 2 it was found that if0 lt 1198770 lt 119877lowast
0 then the disease-free equilibrium point 1198640
is the only one and it also is asymptotically stable in thefeasible region Δ Besides according to Theorem 3 there arenot periodic orbits here As a result all initial condition inΔ which satisfies this interval originates paths tending to1198640 consequently such point must be globally asymptoticallystable
Finally in order to prove the global stability of a uniqueendemic equilibrium 119864lowast in the interior of Δ when 1198770 gt 1 weuse the method of Lyapunov functions This is accomplishedin the next theorem
Theorem5 Assume that1198770 gt 1Then a unique endemic equi-librium 119864lowast = (119868lowast 119877lowast) of system (6) is globally asymptoticallystable in the interior of ΔProof To prove the global asymptotic stability of the uniqueendemic equilibrium 119864lowast we define a Lyapunov function119882(119868 119877) isin R2
0+ 119868 gt 0 119877 gt 0 rarr R given by
119882(119868 119877) = 120583(ln 119868119868lowast + 119868lowast119868 minus 1)+ 120573 (120590 minus 1) (119877 minus 119877lowast minus 119877lowast ln 119877119877lowast )
(26)
Let119882119894 = ln(119868119868lowast) + 119868lowast119868 minus 1 By using 120573119873 minus (120574 + 120583) = 120573119868lowast minus120573(120590 minus 1)119877lowast we have119889119882119894119889119905 = 120573119868lowast (2 minus 119868lowast119868 minus 119868119868lowast)
+ 120573 (120590 minus 1) 119877lowast ( 119877119877lowast minus 1 minus 119868lowast119877119868119877lowast + 119868lowast119868 ) (27)
Let119882119903 = 119877minus119877lowastminus119877lowast ln(119877119877lowast) By using 120574 = 120590120573119877lowast+120583(119877lowast119868lowast)we have
119889119882119903119889119905 = 120590120573119877lowast119868 (2 minus 119877119877lowast minus 119877lowast119877 )+ 120583119877lowast ( 119868119868lowast minus 119877119877lowast minus 119868119877lowast119868lowast119877 + 1) (28)
The derivative of (26) along solution of (6) is given by
119889119882119889119905 = 120583119889119882119894119889119905 + 120573 (120590 minus 1) 119889119882119903119889119905 (29)
By means of (27) and (28) we obtain
119889119882119889119905 = 120583120573 (119868lowast minus (120590 minus 1) 119877lowast) (2 minus 119868lowast119868 minus 119868119868lowast)+ 120573 (120590 minus 1) 120590120573119877lowast119868 (2 minus 119877119877lowast minus 119877lowast119877 )+ 120583120573 (120590 minus 1) 119877lowast (2 minus 119868119877lowast119868lowast119877 minus 119868lowast119877119868119877lowast ) lt 0
(30)
Using
120573119868lowast minus 120573 (120590 minus 1) 119877lowast = 120573119873 minus (120574 + 120583)= (120574 + 120583) (1198770 minus 1) (31)
6 Discrete Dynamics in Nature and Society
we have119889119882119889119905 = 120583 (120574 + 120583) (1198770 minus 1) (2 minus 119868lowast119868 minus 119868119868lowast)+ 120573 (120590 minus 1) 120590120573119877lowast119868 (2 minus 119877119877lowast minus 119877lowast119877 )+ 120583120573 (120590 minus 1) 119877lowast (2 minus 119868119877lowast119868lowast119877 minus 119868lowast119877119868119877lowast ) lt 0
(32)
Therefore if 1198770 gt 1 then 119889119882119889119905 is negative definite Bythe Lyapunov asymptotic stability theorem [17] this resultimplies that119864lowast is globally asymptotically stable in the interiorof ΔRemark 6 Recently Vargas-De-Leon [18] used the Lyapunovfunction119882119894 = ln(119868119868lowast)+119868lowast119868minus1 to prove global stability of thecoexistence equilibrium of two-species mutualism models
4 Case with Social and Nonsocial ContagionRisks of Obesity (120598 gt 0)
41 Equilibria and Nonexistence of Periodic Orbits Wemightfollow the usual method of setting the right-hand side of (4)equal to zero in order to obtain their equilibria This proce-dure is not the most appropriate because it leads to a cubicequation whose analytical solutions are quite complicatedInstead an alternative way to find the equilibrium points isto perform a geometric analysis based on the intersection ofthe nullclines of system (4) The nullclines = 0 (verticaldirections) and = 0 (horizontal directions) are given bythe functions
1198661 (119868) = 1205731198682 minus (120573119873 minus 120598 minus 120583 minus 120574) 119868 minus 120598119873120573 (120590 minus 1) 119868 + 120598 (120590 minus 1) (33)
respectivelyNote that1198661(119868) and1198662(119868) respectively are discontinuous
in 119868 = minus120598120573 and 119868 = minus(120590120598 + 120583)120590120573 Besides it is clear that1198661(119868) rarr +infin (resp 1198662(119868) rarr 120574120590120573) as 119868 rarr +infin and that thefunction1198661(119868) (resp1198662(119868)) is increasing and concave downGiven that 1198661(0) = minus119873(120590 minus 1) lt 0 and 1198662(0) = 0 thenthe functions 1198661(119868) and 1198662(119868) intersect at a single point (theequilibrium point) in the first quadrant (see Figure 3) Thisresult indicates that in Δ an endemic equilibrium point existsand is unique
Regarding whether or not system (4) has periodic orbitswe have precisely the following result
Theorem 7 System (4) does not have periodic orbits in theinterior of ΔProof Consider system (4) for 119868 gt 0 and 119877 gt 0 From theDulac function
Φ (119868 119877) = 1119868 (35)
Equilibrium point
minus120590120576 + 120583
120590120573minus120576
120573
minusN
120590 minus 1
120574
120590120573
R
G1(I)
G2(I)
I
Figure 3 Schematic representation of a branch of the verticalnullcline 1198661(119868) (dashed line) and the horizontal nullcline 1198662(119868)(solid line) Both curves intersect inΔ once at the equilibrium point
Equilibriumpoint
R
(0 N)
R = N minus I
I+ (N 0) I
Figure 4 Trapping region bounded by the dashed line formed bythe horizontal and vertical axes and the diagonal line 119877 = 119873 minus 119868This triangular region contains inside the only equilibrium point
it follows that120597 (Φ119875 (119868 119877))120597119868 + 120597 (Φ119876 (119868 119877))120597119877= minus 1205981198682 (119873 minus (119868 + 119877)) minus (120573 + 120598119868) minus 1205901205981198771198682minus 120590(120573 + 120598119868) minus 120583119868
(36)
Since 120597(Φ119875)120597119868+120597(Φ119876)120597119877 lt 0 inΔ taking into account theDulac criterion we conclude that system (4) has no periodicorbits
42 Local andGlobal Stability In order to determine the localand global stability of the single equilibrium point in Δ weconsider that it is inside of the triangular region bounded bythe dashed line shown in Figure 4 This is a trapping region
Discrete Dynamics in Nature and Society 7
the vector field of system (4) on the boundary point into thebox For the points on the horizontal side = 120574119868 gt 0 and = minus(119868 minus 119868+)(119868 minus 119868minus) where 119868+ and 119868minus are the positive andnegative roots of (120573119868 + 120598)(119868 minus 119873) + (120574 + 120583)119868 respectivelyThus these inequalities imply that the vector field points upand right on the interval (0 119868+) while it goes up and left on(119868+ 119873) Moreover in the vertical side = minus(120590120598 + 120583)119877 le 0and = 120598119873 + 120598(120590 minus 1)119877 gt 0 This implies that the vector fieldis directed downward and right except at the origin where itgoes to the right
Finally on the diagonal side of slope minus1 extending fromthe point (0119873) to (119873 0) it can be shown that
119889119877119889119868 = minus1 + 120583119873(119868 minus 119868lowast+) (119868 minus 119868lowast
minus) (37)
where 119868lowast+and 119868lowastminusare the positive and negative roots of 120590(120573119868 +120598)(119868 minus 119873) + (120574 + 120583)119868 respectively Hence the vector field
for large values of 119868 is almost parallel to the diagonal lineIn a more precise analysis always 119868 minus 119868lowast
minusgt 0 and 119868 minus 119868lowast
+gt0 if 119868 gt 119868lowast
+or 119868 minus 119868lowast
+lt 0 if 119868 lt 119868lowast
+ According to
(37) the first case implies that the vector field points inward(to the right) on the diagonal line because 119889119877119889119868 is morenegative than minus1 while the latter indicates that the vectorfield is directed inside (to the left) the diagonal side because119889119877119889119868 is less negative than minus1 (See Figure 4) Therefore weconclude that the triangular region is effectively a trappingregion
On the other hand Theorem 7 prohibits the existenceof closed orbits inside the triangular region Consequentlythe trajectories entering the triangular region accordingto the Poincare-Bendixson theorem [19] must converge tothe single equilibrium point located inside it Thereforethe equilibrium point inside the triangular region must beasymptotically stable Moreover since this point is the onlypoint of equilibrium in Δ it is also globally asymptoticallystable
The results of this section may be summarized in thefollowing theorem
Theorem 8 System (4) has a unique equilibrium point in thetriangular region Δ which is globally asymptotically stable
5 Numerical Results
In this section we perform a series of numerical simulationsand graphs of system (4) to illustrate the different resultsobtained for each of the two cases of interest 120598 gt 0 and120598 = 0 previously analyzed We will use the parameter valuesreported in the work developed by Ejima et al [8] theirepidemiological meanings and magnitudes are indicated inTable 1
51 Case 120598 = 0 According to the magnitudes of theparameters reported in Table 1 (19) takes the value 119877lowast
0=069363 whereas (21) acquires the value 119868lowast
0= 163304
Table 1 Baseline values and epidemiological meanings of theparameters given by Ejima et al [8]
Parameter Baseline values Epidemiological meaning
119873 100000 individuals(assumed) Population size
1120583 694 (per year) Average life expectancy at birth120573 296 times 10minus7 (peryear)
Transmission rate of obesity dueto social contagion120598 0012 (per year) Nonsocial contagion risk ofobesity120590 80 Relative hazard of obesity amongthe ex-obese1120574 358 (per year) Average duration of obesity
2000
20000
0
4000
4000
6000
600080
00
8000
1000
0
10000
1200
0
12000
1400
0
14000
1600
0
16000
1800
0
18000
2000
0
200002200024000260002800030000
2200
024
000
2600
028
000
3000
0
I (obese)
R (e
x-ob
ese)
Figure 5 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 03 obtained by considering 120573 = 127 times 10minus7Furthermore taking into consideration (10) and (11) theresulting two endemic equilibria are119864lowast
1= (20918 91378) (38)
119864lowast2= (12522 79384) (39)
which as discussed previously are asymptotically stable andunstable respectively
In Figures 5ndash9 we illustrated representative phase por-traits of the dynamic system (6) for some different values of1198770 between 0 and just over 1 according to the bifurcationdiagram shown in Figure 2 These values are obtained bymaintaining fixed in (9) 119873 120583 and 120574 but varying theparameter 120573 Thus if 1198770 takes values between 0 and 119877lowast
0
say 1198770 = 03 we have only one equilibrium point (thedisease-free equilibrium) which is globally asymptoticallystable (see Figure 5) If 1198770 = 119877lowast
0 we have two equilibrium
points one asymptotically stable disease-free equilibriumand the other endemic equilibria point which is nonhyper-bolic (see Figure 6) When 1198770 takes values between 119877lowast0 and
8 Discrete Dynamics in Nature and Society
2000
2000
0
0
4000
4000
6000
6000
8000
800010
000
10000
1200
0
12000
1400
0
14000
1600
0
16000
1800
0
18000
2000
0
20000
2200
0
2400
0I (obese)
R (e
x-ob
ese)
Figure 6 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 119877lowast0 obtained by considering 120573 = 294 times 10minus7
5000
2000
0
0
1000
0
4000
1500
0
6000
2000
0
8000
2500
0
10000
3000
0
12000
3500
0
14000
4000
0
16000
4500
0
18000
5000
0
20000
I (obese)
R (e
x-ob
ese)
Figure 7 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 08 obtained by considering 120573 = 339 times 10minus71 say 1198770 = 08 we have three equilibria one disease-free equilibrium (asymptotically stable) and two endemicequilibria (asymptotically stable and unstable) as shown inFigure 7 If 1198770 = 1 there are two equilibrium pointsone disease-free equilibrium which is nonhyperbolic and theother asymptotically stable endemic equilibrium as shown inFigure 8 Finally if 1198770 takes a slightly greater value to 1 forexample1198770 = 12 we have again two equilibrium points onedisease-free equilibrium (unstable) and the other endemicequilibrium (asymptotically stable) but both are hyperbolic(see Figure 9)
On the other hand in Figures 10ndash14 examples of theevolution in time of 119868(119905) and119877(119905) are presented for the values1198770 = 03 1198770 = 119877lowast
0 1198770 = 08 1198770 = 1 and 1198770 = 12 which
1000
00
2000
0
3000
0
4000
0
5000
0
6000
0
7000
0
8000
0
9000
0
1000
00
1100
00
1200
00
I (obese)
10000
20003000400050006000700080009000
10000110001200013000140001500016000
R (e
x-ob
ese)
Figure 8 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 1 obtained by considering 120573 = 423 times 10minus7
1000
0
10000
0
2000
0
2000
3000
0
30004000
4000
050
000
50006000
6000
07000
7000
0
80009000
8000
090
000
1000011000
1000
00
12000
1100
00
1400013000
1200
0013
0000
1400
00
1500016000
I (obese)
R (e
x-ob
ese)
Figure 9 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 12 obtained by considering 120573 = 508 times 10minus7illustrate respectively some of the dynamics shown in thephase portraits in Figures 5ndash9
52 Case 120598 gt 0 Taking into account the value 120598 =0012 gt 0 given in Table 1 in addition to the othersthere reported nullclines (33) and (34) intersect at the onlyendemic equilibrium point119864lowast = (60890 66805) (40)
which is globally asymptotically stable In Figure 15 thecorresponding phase portrait with several representativetrajectories around this equilibrium point is shown Also
Discrete Dynamics in Nature and Society 9
Time (years)
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 10 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 03 and the initial condition is(119868(0) 119877(0)) = (10000 30000)
Time (years)0 100 200 300
ObeseEx-obese
20000
15000
10000
5000
0
Figure 11 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 119877lowast
0and the initial condition is(119868(0) 119877(0)) = (20000 8000)
performing another series of numerical simulations of system(4) we show for this case in Figure 16 an example of theevolution in time of 119868(119905) and 119877(119905)6 Discussion and Conclusions
According to the results of our analysis of the Ejima et almodel we arrive at the following conclusions It has beenshown that the Ejima et al model [8] if only the socialcontagion (120598 = 0) is considered presents the two bifurcationvalues 1198770 = 119877lowast
0and 1198770 = 1 which results in what is
known as backward bifurcation From a mathematical pointof view the backward bifurcation is apparently constitutedby the combination of two different kinds of bifurcations
Time (years)
40000
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 12 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 08 and the initial condition is(119868(0) 119877(0)) = (30000 12000)
Time (years)
40000
50000
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 13 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 1 and the initial condition is(119868(0) 119877(0)) = (18000 2500)a saddle-node bifurcation that occurs at 1198770 = 119877lowast
0 whose
upper (asymptotically stable) and lower (unstable) quadraticbranches are defined for 119877lowast
0lt 1198770 and a transcritical
bifurcation that takes place in 1198770 = 1 whose horizontalbranches on the 1198770 axis are asymptotically stable for 1198770 lt 1and unstable for 1 lt 1198770 For values of 1 lt 1198770 only theupper quadratic branch asymptotically stable of the saddle-node bifurcation and the unstable horizontal branch of thetranscritical bifurcation have ameaning in the feasible regionΔ The diagram shown in Figure 2 suggests the presenceof these two bifurcations in the vicinities of the points(119877lowast0 119868lowast0) and (1 0) Of course we should go beyond the
graphic evidence and employ the qualitative analysis of localbifurcations in those points to verify our statement even
10 Discrete Dynamics in Nature and Society
Time (years)0 100 200 300
ObeseEx-obese
140000
120000
100000
80000
60000
40000
20000
0
Figure 14 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 12 and the initial condition is(119868(0) 119877(0)) = (100000 80000)
1000
0
10000
0
2000
0
2000
3000
0
30004000
4000
0
5000
0
50006000
6000
0
7000
7000
0
80009000
8000
0
9000
0
1000011000
1000
00
12000
1100
00
1400013000
1200
00
1300
00
1400
00
1500016000
I (obese)
R (e
x-ob
ese)
Figure 15 Phase portrait for system (4) for the case 120598 = 0012 gt 0(considering 120573 = 296 times 10minus7)without the realization of this depth analysis our discussionsand conclusions can be supported firmly from the resultsshown in this work of investigation
Indeed the presence of the different branches of familiesof equilibrium points shown in Figure 2 causes the system todisplay as the parameter1198770 is varied a bistable region and asa consequence the hysteresis phenomenon In order to showin more detail the presence of hysteresis in the bifurcationdiagram mentioned before suppose we start the system inthe state119860 (see Figure 17) and then slowly increase 1198770 (whichis indicated by an arrow to the right just beneath the axis1198770 of Figure 17) We remain at the origin until 1198770 = 1when the origin loses its stability At this point at the slightestdisturbance the system will jump to the state 119861 located in the
0
20000
40000
60000
80000
100000
120000
Time (years)0 100 200 300
ObeseEx-obese
Figure 16 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0012 gt 0 and the initial condition is(100000 50000)
B
A
1
C
80000
70000
60000
50000
40000
30000
20000
10000
0
Infe
cted
(I)
Rlowast0
Basic reproductive number (R0)
Figure 17 Schematic representation of hysteresis phenomenon thatexhibits the bifurcation diagram of system (4) for the case 120598 = 0Here the forward and backward paths are not equal as the parameter1198770 is changedstable upper branch Insofar as1198770 increases its value (1198770 gt 1)the equilibrium points of the system will move along thisbranch away from the state119861 If now1198770 continuously reducedits value equilibrium states will return to the state 119861 cross itand reach the state 119862 (in which 1198770 = 119877lowast
0) At this point at
the slightest change again equilibrium states will jump backto the origin (in 1198770 = 119877lowast0 ) and if 1198770 continues to decrease wewill return to the state119860 Consequently the system exhibits alack of reversibility or it is said that it has memory (or that itpresents hysteresis) because the forward and backward pathsare not identical
From an epidemiological point of view in this backwardbifurcation if 0 lt 1198770 lt 119877lowast
0 no matter where the initial
Discrete Dynamics in Nature and Society 11
conditions are taken solutions always tend to the straightline of stable disease-free points (that correspond to a never-obese population) When 119877lowast
0lt 1198770 lt 1 if the initial
conditions are located below the unstable quadratic branchof endemic equilibrium points (obese people) then solutionscontinue going to the horizontal line of stable disease-freepoints otherwise they will be directed to the stable curveof endemic points located at the upper quadratic branchFinally if 1 lt 1198770 no matter again where the initialconditions are situated solutions will continue heading forthe stable endemic points of such upper branch Should benoted that unlike what happens in systems that exhibit thephenomenon of forward bifurcation in which the endemicequilibrium exists only for 1198770 gt 1 [20] in systems exhibitinga backward bifurcation under certain initial conditions itis possible to have endemic states when 1198770 lt 1 In ourcase the endemic equilibrium that exists for 1198770 just aboveone has a large obese population so the result of 1198770 risingabove one would be a sudden and dramatic jump in thenumber of infective Moreover reducing 1198770 back below onewould not eradicate the obesity if the size of the infectedpopulation is greater or equal than 119868lowast
0 the corresponding
value of 119877lowast0 In order to eradicate the disease 1198770 must be
further reduced so it just needs to be slightly less than119877lowast0 to suddenly enter the region where only stable disease-
free equilibrium (never-obese population) exists Both suchdifferent abrupt behaviours are the result of hysteresisphenomenon
It can be seen from the bifurcation diagram shown inFigure 2 that theway inwhich obesity infected population canbe lowered (or raised) is through the reduction (or increase)of 1198770 parameter which in accordance with (9) depends onthe four quantities 119873 120573 120583 and 120574 Since we have assumedin (1) that the natural death and birth rates are equal anddenoted by 120583 the population size119873 is constantTherefore1198770can only change according to the following forms Firstly 1198770diminishes if the natural rate 120583 (of death or birth) increasesthe transmission rate 120573 decreases or the rate 120574 grows atwhich the infected individuals become recovered On theother hand 1198770 increment its value if the opposite occurs 120583decreases 120573 goes up or 120574 declines It should be noted thatchanges in 120583 could be slower and achieved in the long termprobably in decades which is not a viable option Converselychanges in 120573 or 120574 might be faster and perhaps occur infewer years Since the focus is on reducing obesity levels andtherefore the values of 1198770 it is of interest to seek mechanismsto decrease120573 andor increase 120574The former could be achievedby sensitizing at the population in general about the benefitsassociatedwith having a healthy and balanced diet and also bydesigning effective health and informative public preventionpoliciesThe latter could be done by providing better medicalassistance and making it available to the obese populationIt should be emphasized that this last strategy is not feasiblein countries with high levels of obesity as Mexico because atthe present time it is very expensive and will become evenmore expensive as the affected population increases In thisregard we believe that it is much more convenient that thegovernments of the affected countries implement broaderand more effective preventive policies
It is worth mentioning that in Mexico there have beenstudies to quantify more precisely the prevalence of peoplesuffering from overweight and obesity It is estimated that713 of Mexican adults aged 20 or more are found inthis condition [21] While trends show slowing down of theincrease in the obesity prevalence there is no evidence toinfer that prevalence will decrease in the coming years Inthis sense studies like this allow us to analyze the trends ofthe population with obesity and study the different possiblescenarios Particularly with the model studied this couldbe done for Mexico if we knew accurately the values ofits parameters 120573 and 120574 To our knowledge there is notsuch information Nevertheless these parameters could beestimated in a short or medium term based on statisti-cal information collected on obesity in our country Theknowledge of how these parameters behave over time wouldcontribute in quantifying the effectiveness of the Mexicanpublic policies against obesity
With respect to the model studied it is pertinent tocomment about the term of relapse 120590[120573119868(119905) + 120598]119877(119905) of thesecond equation (1) In the first place in this work it has beenconsidered that the parameter 120590 the relative risk of weightregain among ex-obese individuals is greater than unity (120590 gt1)This condition represents the high risk that recovered (ex-obese) individuals will become obese again However it isalso plausible to consider that 0 lt 120590 lt 1 which representsthat ex-obese population is more resistant to being obeseagain that is they are more aware of the health risks thatthis would represent The consideration that 0 lt 120590 lt 1 doesnot change the results of the stability analysis and periodicorbits performed in Sections 3 and 4 With respect to thelatter the nullcline 1198661 (given by (33)) changes to 1198661015840
1 where1198661015840
1= minus1198661 while 1198662 (34) remains the same The intersection
of the nullclines 11986610158401and 1198662 again occurs at only one point
in the region Δ and its stability is the same Regarding thisaspect we could conclude that regardless of the value ofthe parameter 120590 there is always the possibility that the ex-obese population is at risk of relapse and becoming obeseagain of course this possibility is less if 0 lt 120590 lt 1 andgreater if 120590 gt 1 In this sense public health programs againstobesity designed by governments may be more effectiveif they take into account the awareness of the ex-obesepopulation
Finally with regard to the mentioned relapse term120590[120573119868(119905) + 120598]119877(119905) if we consider only the effects of socialcontagion (which corresponds to 120598 = 0) it is reduced tothe nonlinear term 120590120573119868(119905)119877(119905) which causes system (4) toexhibit a backward bifurcation If the effects of nonsocialcontagion (that is 120598 gt 0) are also incorporated the presenceof the linear term 120590120573120598119877(119905) causes the backward bifurcationto disappear in system (4) and in its place it only presentsan equilibrium point global asymptotically stable The lattercase suggests that the way in which the effects of nonsocialcontagion are introduced into the model by means of thelinear part of the relapse term should be modified in orderto maintain much of the interesting dynamic that is presentin the backward bifurcation Certainly an alternative wayis considering instead of a linear a nonlinear contribution
12 Discrete Dynamics in Nature and Society
Perhaps it is possible to build amore general nonlinearmodelwhere in the case in which the contribution due to nonsocialeffects gradually fades the backward bifurcation of the Ejimaet al model could be obtained as a limit case
Competing Interests
The authors declare that they have no conflict of interests
Acknowledgments
This research was supported by the UACM-SECITI ResearchGrant PI-2013-34 Cruz Vargas-De-Leon would like toexpress his indebtedness to Dr Ramon Reyes Carreto for hishospitality during his stay as visiting professor at the UAGroin Chilpancingo
References
[1] World Health Organization Obesity Preventing and Managingthe Global Epidemic Report of a WHO Consultation WHOTechnical Report Series No 894 WHO Geneva Switzerland2000 httpswwwgooglecommxurlsa=tamprct=jampq=ampesrc=sampsource=webampcd=1ampved=0ahUKEwjlno 3tsDRAhVqi1QKH-W0rBPIQFggZMAAampurl=http3A2F2Fwhqlibdocwhoint2Ftrs2FWHO TRS 894pdfampusg=AFQjCNFLusNXrJz-xFKX1xpQwkHLEr-Lohgampbvm=bv144224172damcampcad=rja
[2] C L Vanvrancken-Tompkins andMaM Sothern ldquoPrevencionde la obesidad en los ninos desde el nacimiento hasta loscincoanosrdquo in Enciclopedia Sobre el Desarrollo de la PrimeraInfancia R Tremblay R Barr R Peters and M Boivin Edsvol 1ndash7 Centre of Excellence for EarlyChildhoodDevelopmentQuebec Canada 2010
[3] S K Garg H Maurer K Reed and R Selagamsetty ldquoDiabetesand cancer two diseases with obesity as a common risk factorrdquoDiabetes Obesity andMetabolism vol 16 no 2 pp 97ndash110 2014
[4] National Heart ldquoLung and Blood Institute World Health WhatAre the Health Risks of Overweight and Obesityrdquo httpswwwnhlbinihgovhealthhealth-topicstopicsoberisks
[5] World Health Organization (WHO) Global Health Observa-tory Data Repository WHO Geneva Switzerland 2011 httpappswhointghodata
[6] World Health Organization The Global Strategy on Diet Phys-ical Activity and Health (DPAS) WHO Geneva Switzerland2004 httpwwwwhointnmhwha59dpasen
[7] N A Christakis and J H Fowler ldquoThe spread of obesity in alarge social network over 32 yearsrdquoThe New England Journal ofMedicine vol 357 no 4 pp 370ndash379 2007
[8] K Ejima K Aihara and H Nishiura ldquoModeling the obesityepidemic social contagion and its implications for controlrdquoTheoretical Biology and Medical Modelling vol 10 article 17 pp1ndash13 2013
[9] A J Arenas G Gonzalez-Parra and L Jodar ldquoPeriodic solu-tions of nonautonomous differential systems modeling obesitypopulationrdquo Chaos Solitons amp Fractals vol 42 no 2 pp 1234ndash1244 2009
[10] G Gonzalez-Parra L Acedo R-J Villanueva Mico and A JArenas ldquoModeling the social obesity epidemic with stochasticnetworksrdquo Physica A Statistical Mechanics and its Applicationsvol 389 no 17 pp 3692ndash3701 2010
[11] G Gonzalez-Parra L Jodar F J Santonja and R J VillanuevaldquoAn age-structured model for childhood obesity mathematicalpopulation studiesrdquo An International Journal of MathematicalDemography vol 17 no 1 pp 1ndash11 2010
[12] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 Article ID e1000968 2010
[13] L Jodar F J Santonja and G Gonzalez-Parra ldquoModelingdynamics of infant obesity in the region of Valencia SpainrdquoComputers andMathematics with Applications vol 56 no 3 pp679ndash689 2008
[14] F-J Santonja R-J Villanueva L Jodar and G Gonzalez-Parra ldquoMathematical modelling of social obesity epidemic inthe region of Valencia Spainrdquo Mathematical and ComputerModelling of Dynamical Systems vol 16 no 1 pp 23ndash34 2010
[15] F J Santonja and L Shaikhet ldquoProbabilistic stability analysisof social obesity epidemic by a delayed stochastic modelrdquoNonlinear Analysis Real World Applications vol 17 pp 114ndash1252014
[16] D Tudor ldquoA deterministic model for herpes infections inhuman and animal populationsrdquo SIAM Review vol 32 no 1pp 136ndash139 1990
[17] A M Lyapunov The general problem of the stability of motionTaylor amp Francis London England 1992
[18] C Vargas-De-Leon ldquoLyapunov functions for two-species coop-erative systemsrdquo Applied Mathematics and Computation vol219 no 5 pp 2493ndash2497 2012
[19] D W Jordan and P Smith Nonlinear Ordinary DifferentialEquations An Introduction for Scientists and Engineers OxfordUniversity Press New York NY USA 4th edition 2007
[20] C M Kribs-Zaleta and J X Velasco-Hernandez ldquoA simplevaccination model with multiple endemic statesrdquoMathematicalBiosciences vol 164 no 2 pp 183ndash201 2000
[21] S Barquera I Campos-Nonato L Hernandez-Barrera APedroza-Tobıas and J A Rivera-Dommarco ldquoPrevalence ofobesity in mexican adults ensanut 2012rdquo Salud Publica deMexico vol 55 supplement 2 pp S151ndashS160 2013
we have119889119882119889119905 = 120583 (120574 + 120583) (1198770 minus 1) (2 minus 119868lowast119868 minus 119868119868lowast)+ 120573 (120590 minus 1) 120590120573119877lowast119868 (2 minus 119877119877lowast minus 119877lowast119877 )+ 120583120573 (120590 minus 1) 119877lowast (2 minus 119868119877lowast119868lowast119877 minus 119868lowast119877119868119877lowast ) lt 0
(32)
Therefore if 1198770 gt 1 then 119889119882119889119905 is negative definite Bythe Lyapunov asymptotic stability theorem [17] this resultimplies that119864lowast is globally asymptotically stable in the interiorof ΔRemark 6 Recently Vargas-De-Leon [18] used the Lyapunovfunction119882119894 = ln(119868119868lowast)+119868lowast119868minus1 to prove global stability of thecoexistence equilibrium of two-species mutualism models
4 Case with Social and Nonsocial ContagionRisks of Obesity (120598 gt 0)
41 Equilibria and Nonexistence of Periodic Orbits Wemightfollow the usual method of setting the right-hand side of (4)equal to zero in order to obtain their equilibria This proce-dure is not the most appropriate because it leads to a cubicequation whose analytical solutions are quite complicatedInstead an alternative way to find the equilibrium points isto perform a geometric analysis based on the intersection ofthe nullclines of system (4) The nullclines = 0 (verticaldirections) and = 0 (horizontal directions) are given bythe functions
1198661 (119868) = 1205731198682 minus (120573119873 minus 120598 minus 120583 minus 120574) 119868 minus 120598119873120573 (120590 minus 1) 119868 + 120598 (120590 minus 1) (33)
respectivelyNote that1198661(119868) and1198662(119868) respectively are discontinuous
in 119868 = minus120598120573 and 119868 = minus(120590120598 + 120583)120590120573 Besides it is clear that1198661(119868) rarr +infin (resp 1198662(119868) rarr 120574120590120573) as 119868 rarr +infin and that thefunction1198661(119868) (resp1198662(119868)) is increasing and concave downGiven that 1198661(0) = minus119873(120590 minus 1) lt 0 and 1198662(0) = 0 thenthe functions 1198661(119868) and 1198662(119868) intersect at a single point (theequilibrium point) in the first quadrant (see Figure 3) Thisresult indicates that in Δ an endemic equilibrium point existsand is unique
Regarding whether or not system (4) has periodic orbitswe have precisely the following result
Theorem 7 System (4) does not have periodic orbits in theinterior of ΔProof Consider system (4) for 119868 gt 0 and 119877 gt 0 From theDulac function
Φ (119868 119877) = 1119868 (35)
Equilibrium point
minus120590120576 + 120583
120590120573minus120576
120573
minusN
120590 minus 1
120574
120590120573
R
G1(I)
G2(I)
I
Figure 3 Schematic representation of a branch of the verticalnullcline 1198661(119868) (dashed line) and the horizontal nullcline 1198662(119868)(solid line) Both curves intersect inΔ once at the equilibrium point
Equilibriumpoint
R
(0 N)
R = N minus I
I+ (N 0) I
Figure 4 Trapping region bounded by the dashed line formed bythe horizontal and vertical axes and the diagonal line 119877 = 119873 minus 119868This triangular region contains inside the only equilibrium point
it follows that120597 (Φ119875 (119868 119877))120597119868 + 120597 (Φ119876 (119868 119877))120597119877= minus 1205981198682 (119873 minus (119868 + 119877)) minus (120573 + 120598119868) minus 1205901205981198771198682minus 120590(120573 + 120598119868) minus 120583119868
(36)
Since 120597(Φ119875)120597119868+120597(Φ119876)120597119877 lt 0 inΔ taking into account theDulac criterion we conclude that system (4) has no periodicorbits
42 Local andGlobal Stability In order to determine the localand global stability of the single equilibrium point in Δ weconsider that it is inside of the triangular region bounded bythe dashed line shown in Figure 4 This is a trapping region
Discrete Dynamics in Nature and Society 7
the vector field of system (4) on the boundary point into thebox For the points on the horizontal side = 120574119868 gt 0 and = minus(119868 minus 119868+)(119868 minus 119868minus) where 119868+ and 119868minus are the positive andnegative roots of (120573119868 + 120598)(119868 minus 119873) + (120574 + 120583)119868 respectivelyThus these inequalities imply that the vector field points upand right on the interval (0 119868+) while it goes up and left on(119868+ 119873) Moreover in the vertical side = minus(120590120598 + 120583)119877 le 0and = 120598119873 + 120598(120590 minus 1)119877 gt 0 This implies that the vector fieldis directed downward and right except at the origin where itgoes to the right
Finally on the diagonal side of slope minus1 extending fromthe point (0119873) to (119873 0) it can be shown that
119889119877119889119868 = minus1 + 120583119873(119868 minus 119868lowast+) (119868 minus 119868lowast
minus) (37)
where 119868lowast+and 119868lowastminusare the positive and negative roots of 120590(120573119868 +120598)(119868 minus 119873) + (120574 + 120583)119868 respectively Hence the vector field
for large values of 119868 is almost parallel to the diagonal lineIn a more precise analysis always 119868 minus 119868lowast
minusgt 0 and 119868 minus 119868lowast
+gt0 if 119868 gt 119868lowast
+or 119868 minus 119868lowast
+lt 0 if 119868 lt 119868lowast
+ According to
(37) the first case implies that the vector field points inward(to the right) on the diagonal line because 119889119877119889119868 is morenegative than minus1 while the latter indicates that the vectorfield is directed inside (to the left) the diagonal side because119889119877119889119868 is less negative than minus1 (See Figure 4) Therefore weconclude that the triangular region is effectively a trappingregion
On the other hand Theorem 7 prohibits the existenceof closed orbits inside the triangular region Consequentlythe trajectories entering the triangular region accordingto the Poincare-Bendixson theorem [19] must converge tothe single equilibrium point located inside it Thereforethe equilibrium point inside the triangular region must beasymptotically stable Moreover since this point is the onlypoint of equilibrium in Δ it is also globally asymptoticallystable
The results of this section may be summarized in thefollowing theorem
Theorem 8 System (4) has a unique equilibrium point in thetriangular region Δ which is globally asymptotically stable
5 Numerical Results
In this section we perform a series of numerical simulationsand graphs of system (4) to illustrate the different resultsobtained for each of the two cases of interest 120598 gt 0 and120598 = 0 previously analyzed We will use the parameter valuesreported in the work developed by Ejima et al [8] theirepidemiological meanings and magnitudes are indicated inTable 1
51 Case 120598 = 0 According to the magnitudes of theparameters reported in Table 1 (19) takes the value 119877lowast
0=069363 whereas (21) acquires the value 119868lowast
0= 163304
Table 1 Baseline values and epidemiological meanings of theparameters given by Ejima et al [8]
Parameter Baseline values Epidemiological meaning
119873 100000 individuals(assumed) Population size
1120583 694 (per year) Average life expectancy at birth120573 296 times 10minus7 (peryear)
Transmission rate of obesity dueto social contagion120598 0012 (per year) Nonsocial contagion risk ofobesity120590 80 Relative hazard of obesity amongthe ex-obese1120574 358 (per year) Average duration of obesity
2000
20000
0
4000
4000
6000
600080
00
8000
1000
0
10000
1200
0
12000
1400
0
14000
1600
0
16000
1800
0
18000
2000
0
200002200024000260002800030000
2200
024
000
2600
028
000
3000
0
I (obese)
R (e
x-ob
ese)
Figure 5 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 03 obtained by considering 120573 = 127 times 10minus7Furthermore taking into consideration (10) and (11) theresulting two endemic equilibria are119864lowast
1= (20918 91378) (38)
119864lowast2= (12522 79384) (39)
which as discussed previously are asymptotically stable andunstable respectively
In Figures 5ndash9 we illustrated representative phase por-traits of the dynamic system (6) for some different values of1198770 between 0 and just over 1 according to the bifurcationdiagram shown in Figure 2 These values are obtained bymaintaining fixed in (9) 119873 120583 and 120574 but varying theparameter 120573 Thus if 1198770 takes values between 0 and 119877lowast
0
say 1198770 = 03 we have only one equilibrium point (thedisease-free equilibrium) which is globally asymptoticallystable (see Figure 5) If 1198770 = 119877lowast
0 we have two equilibrium
points one asymptotically stable disease-free equilibriumand the other endemic equilibria point which is nonhyper-bolic (see Figure 6) When 1198770 takes values between 119877lowast0 and
8 Discrete Dynamics in Nature and Society
2000
2000
0
0
4000
4000
6000
6000
8000
800010
000
10000
1200
0
12000
1400
0
14000
1600
0
16000
1800
0
18000
2000
0
20000
2200
0
2400
0I (obese)
R (e
x-ob
ese)
Figure 6 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 119877lowast0 obtained by considering 120573 = 294 times 10minus7
5000
2000
0
0
1000
0
4000
1500
0
6000
2000
0
8000
2500
0
10000
3000
0
12000
3500
0
14000
4000
0
16000
4500
0
18000
5000
0
20000
I (obese)
R (e
x-ob
ese)
Figure 7 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 08 obtained by considering 120573 = 339 times 10minus71 say 1198770 = 08 we have three equilibria one disease-free equilibrium (asymptotically stable) and two endemicequilibria (asymptotically stable and unstable) as shown inFigure 7 If 1198770 = 1 there are two equilibrium pointsone disease-free equilibrium which is nonhyperbolic and theother asymptotically stable endemic equilibrium as shown inFigure 8 Finally if 1198770 takes a slightly greater value to 1 forexample1198770 = 12 we have again two equilibrium points onedisease-free equilibrium (unstable) and the other endemicequilibrium (asymptotically stable) but both are hyperbolic(see Figure 9)
On the other hand in Figures 10ndash14 examples of theevolution in time of 119868(119905) and119877(119905) are presented for the values1198770 = 03 1198770 = 119877lowast
0 1198770 = 08 1198770 = 1 and 1198770 = 12 which
1000
00
2000
0
3000
0
4000
0
5000
0
6000
0
7000
0
8000
0
9000
0
1000
00
1100
00
1200
00
I (obese)
10000
20003000400050006000700080009000
10000110001200013000140001500016000
R (e
x-ob
ese)
Figure 8 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 1 obtained by considering 120573 = 423 times 10minus7
1000
0
10000
0
2000
0
2000
3000
0
30004000
4000
050
000
50006000
6000
07000
7000
0
80009000
8000
090
000
1000011000
1000
00
12000
1100
00
1400013000
1200
0013
0000
1400
00
1500016000
I (obese)
R (e
x-ob
ese)
Figure 9 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 12 obtained by considering 120573 = 508 times 10minus7illustrate respectively some of the dynamics shown in thephase portraits in Figures 5ndash9
52 Case 120598 gt 0 Taking into account the value 120598 =0012 gt 0 given in Table 1 in addition to the othersthere reported nullclines (33) and (34) intersect at the onlyendemic equilibrium point119864lowast = (60890 66805) (40)
which is globally asymptotically stable In Figure 15 thecorresponding phase portrait with several representativetrajectories around this equilibrium point is shown Also
Discrete Dynamics in Nature and Society 9
Time (years)
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 10 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 03 and the initial condition is(119868(0) 119877(0)) = (10000 30000)
Time (years)0 100 200 300
ObeseEx-obese
20000
15000
10000
5000
0
Figure 11 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 119877lowast
0and the initial condition is(119868(0) 119877(0)) = (20000 8000)
performing another series of numerical simulations of system(4) we show for this case in Figure 16 an example of theevolution in time of 119868(119905) and 119877(119905)6 Discussion and Conclusions
According to the results of our analysis of the Ejima et almodel we arrive at the following conclusions It has beenshown that the Ejima et al model [8] if only the socialcontagion (120598 = 0) is considered presents the two bifurcationvalues 1198770 = 119877lowast
0and 1198770 = 1 which results in what is
known as backward bifurcation From a mathematical pointof view the backward bifurcation is apparently constitutedby the combination of two different kinds of bifurcations
Time (years)
40000
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 12 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 08 and the initial condition is(119868(0) 119877(0)) = (30000 12000)
Time (years)
40000
50000
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 13 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 1 and the initial condition is(119868(0) 119877(0)) = (18000 2500)a saddle-node bifurcation that occurs at 1198770 = 119877lowast
0 whose
upper (asymptotically stable) and lower (unstable) quadraticbranches are defined for 119877lowast
0lt 1198770 and a transcritical
bifurcation that takes place in 1198770 = 1 whose horizontalbranches on the 1198770 axis are asymptotically stable for 1198770 lt 1and unstable for 1 lt 1198770 For values of 1 lt 1198770 only theupper quadratic branch asymptotically stable of the saddle-node bifurcation and the unstable horizontal branch of thetranscritical bifurcation have ameaning in the feasible regionΔ The diagram shown in Figure 2 suggests the presenceof these two bifurcations in the vicinities of the points(119877lowast0 119868lowast0) and (1 0) Of course we should go beyond the
graphic evidence and employ the qualitative analysis of localbifurcations in those points to verify our statement even
10 Discrete Dynamics in Nature and Society
Time (years)0 100 200 300
ObeseEx-obese
140000
120000
100000
80000
60000
40000
20000
0
Figure 14 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 12 and the initial condition is(119868(0) 119877(0)) = (100000 80000)
1000
0
10000
0
2000
0
2000
3000
0
30004000
4000
0
5000
0
50006000
6000
0
7000
7000
0
80009000
8000
0
9000
0
1000011000
1000
00
12000
1100
00
1400013000
1200
00
1300
00
1400
00
1500016000
I (obese)
R (e
x-ob
ese)
Figure 15 Phase portrait for system (4) for the case 120598 = 0012 gt 0(considering 120573 = 296 times 10minus7)without the realization of this depth analysis our discussionsand conclusions can be supported firmly from the resultsshown in this work of investigation
Indeed the presence of the different branches of familiesof equilibrium points shown in Figure 2 causes the system todisplay as the parameter1198770 is varied a bistable region and asa consequence the hysteresis phenomenon In order to showin more detail the presence of hysteresis in the bifurcationdiagram mentioned before suppose we start the system inthe state119860 (see Figure 17) and then slowly increase 1198770 (whichis indicated by an arrow to the right just beneath the axis1198770 of Figure 17) We remain at the origin until 1198770 = 1when the origin loses its stability At this point at the slightestdisturbance the system will jump to the state 119861 located in the
0
20000
40000
60000
80000
100000
120000
Time (years)0 100 200 300
ObeseEx-obese
Figure 16 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0012 gt 0 and the initial condition is(100000 50000)
B
A
1
C
80000
70000
60000
50000
40000
30000
20000
10000
0
Infe
cted
(I)
Rlowast0
Basic reproductive number (R0)
Figure 17 Schematic representation of hysteresis phenomenon thatexhibits the bifurcation diagram of system (4) for the case 120598 = 0Here the forward and backward paths are not equal as the parameter1198770 is changedstable upper branch Insofar as1198770 increases its value (1198770 gt 1)the equilibrium points of the system will move along thisbranch away from the state119861 If now1198770 continuously reducedits value equilibrium states will return to the state 119861 cross itand reach the state 119862 (in which 1198770 = 119877lowast
0) At this point at
the slightest change again equilibrium states will jump backto the origin (in 1198770 = 119877lowast0 ) and if 1198770 continues to decrease wewill return to the state119860 Consequently the system exhibits alack of reversibility or it is said that it has memory (or that itpresents hysteresis) because the forward and backward pathsare not identical
From an epidemiological point of view in this backwardbifurcation if 0 lt 1198770 lt 119877lowast
0 no matter where the initial
Discrete Dynamics in Nature and Society 11
conditions are taken solutions always tend to the straightline of stable disease-free points (that correspond to a never-obese population) When 119877lowast
0lt 1198770 lt 1 if the initial
conditions are located below the unstable quadratic branchof endemic equilibrium points (obese people) then solutionscontinue going to the horizontal line of stable disease-freepoints otherwise they will be directed to the stable curveof endemic points located at the upper quadratic branchFinally if 1 lt 1198770 no matter again where the initialconditions are situated solutions will continue heading forthe stable endemic points of such upper branch Should benoted that unlike what happens in systems that exhibit thephenomenon of forward bifurcation in which the endemicequilibrium exists only for 1198770 gt 1 [20] in systems exhibitinga backward bifurcation under certain initial conditions itis possible to have endemic states when 1198770 lt 1 In ourcase the endemic equilibrium that exists for 1198770 just aboveone has a large obese population so the result of 1198770 risingabove one would be a sudden and dramatic jump in thenumber of infective Moreover reducing 1198770 back below onewould not eradicate the obesity if the size of the infectedpopulation is greater or equal than 119868lowast
0 the corresponding
value of 119877lowast0 In order to eradicate the disease 1198770 must be
further reduced so it just needs to be slightly less than119877lowast0 to suddenly enter the region where only stable disease-
free equilibrium (never-obese population) exists Both suchdifferent abrupt behaviours are the result of hysteresisphenomenon
It can be seen from the bifurcation diagram shown inFigure 2 that theway inwhich obesity infected population canbe lowered (or raised) is through the reduction (or increase)of 1198770 parameter which in accordance with (9) depends onthe four quantities 119873 120573 120583 and 120574 Since we have assumedin (1) that the natural death and birth rates are equal anddenoted by 120583 the population size119873 is constantTherefore1198770can only change according to the following forms Firstly 1198770diminishes if the natural rate 120583 (of death or birth) increasesthe transmission rate 120573 decreases or the rate 120574 grows atwhich the infected individuals become recovered On theother hand 1198770 increment its value if the opposite occurs 120583decreases 120573 goes up or 120574 declines It should be noted thatchanges in 120583 could be slower and achieved in the long termprobably in decades which is not a viable option Converselychanges in 120573 or 120574 might be faster and perhaps occur infewer years Since the focus is on reducing obesity levels andtherefore the values of 1198770 it is of interest to seek mechanismsto decrease120573 andor increase 120574The former could be achievedby sensitizing at the population in general about the benefitsassociatedwith having a healthy and balanced diet and also bydesigning effective health and informative public preventionpoliciesThe latter could be done by providing better medicalassistance and making it available to the obese populationIt should be emphasized that this last strategy is not feasiblein countries with high levels of obesity as Mexico because atthe present time it is very expensive and will become evenmore expensive as the affected population increases In thisregard we believe that it is much more convenient that thegovernments of the affected countries implement broaderand more effective preventive policies
It is worth mentioning that in Mexico there have beenstudies to quantify more precisely the prevalence of peoplesuffering from overweight and obesity It is estimated that713 of Mexican adults aged 20 or more are found inthis condition [21] While trends show slowing down of theincrease in the obesity prevalence there is no evidence toinfer that prevalence will decrease in the coming years Inthis sense studies like this allow us to analyze the trends ofthe population with obesity and study the different possiblescenarios Particularly with the model studied this couldbe done for Mexico if we knew accurately the values ofits parameters 120573 and 120574 To our knowledge there is notsuch information Nevertheless these parameters could beestimated in a short or medium term based on statisti-cal information collected on obesity in our country Theknowledge of how these parameters behave over time wouldcontribute in quantifying the effectiveness of the Mexicanpublic policies against obesity
With respect to the model studied it is pertinent tocomment about the term of relapse 120590[120573119868(119905) + 120598]119877(119905) of thesecond equation (1) In the first place in this work it has beenconsidered that the parameter 120590 the relative risk of weightregain among ex-obese individuals is greater than unity (120590 gt1)This condition represents the high risk that recovered (ex-obese) individuals will become obese again However it isalso plausible to consider that 0 lt 120590 lt 1 which representsthat ex-obese population is more resistant to being obeseagain that is they are more aware of the health risks thatthis would represent The consideration that 0 lt 120590 lt 1 doesnot change the results of the stability analysis and periodicorbits performed in Sections 3 and 4 With respect to thelatter the nullcline 1198661 (given by (33)) changes to 1198661015840
1 where1198661015840
1= minus1198661 while 1198662 (34) remains the same The intersection
of the nullclines 11986610158401and 1198662 again occurs at only one point
in the region Δ and its stability is the same Regarding thisaspect we could conclude that regardless of the value ofthe parameter 120590 there is always the possibility that the ex-obese population is at risk of relapse and becoming obeseagain of course this possibility is less if 0 lt 120590 lt 1 andgreater if 120590 gt 1 In this sense public health programs againstobesity designed by governments may be more effectiveif they take into account the awareness of the ex-obesepopulation
Finally with regard to the mentioned relapse term120590[120573119868(119905) + 120598]119877(119905) if we consider only the effects of socialcontagion (which corresponds to 120598 = 0) it is reduced tothe nonlinear term 120590120573119868(119905)119877(119905) which causes system (4) toexhibit a backward bifurcation If the effects of nonsocialcontagion (that is 120598 gt 0) are also incorporated the presenceof the linear term 120590120573120598119877(119905) causes the backward bifurcationto disappear in system (4) and in its place it only presentsan equilibrium point global asymptotically stable The lattercase suggests that the way in which the effects of nonsocialcontagion are introduced into the model by means of thelinear part of the relapse term should be modified in orderto maintain much of the interesting dynamic that is presentin the backward bifurcation Certainly an alternative wayis considering instead of a linear a nonlinear contribution
12 Discrete Dynamics in Nature and Society
Perhaps it is possible to build amore general nonlinearmodelwhere in the case in which the contribution due to nonsocialeffects gradually fades the backward bifurcation of the Ejimaet al model could be obtained as a limit case
Competing Interests
The authors declare that they have no conflict of interests
Acknowledgments
This research was supported by the UACM-SECITI ResearchGrant PI-2013-34 Cruz Vargas-De-Leon would like toexpress his indebtedness to Dr Ramon Reyes Carreto for hishospitality during his stay as visiting professor at the UAGroin Chilpancingo
References
[1] World Health Organization Obesity Preventing and Managingthe Global Epidemic Report of a WHO Consultation WHOTechnical Report Series No 894 WHO Geneva Switzerland2000 httpswwwgooglecommxurlsa=tamprct=jampq=ampesrc=sampsource=webampcd=1ampved=0ahUKEwjlno 3tsDRAhVqi1QKH-W0rBPIQFggZMAAampurl=http3A2F2Fwhqlibdocwhoint2Ftrs2FWHO TRS 894pdfampusg=AFQjCNFLusNXrJz-xFKX1xpQwkHLEr-Lohgampbvm=bv144224172damcampcad=rja
[2] C L Vanvrancken-Tompkins andMaM Sothern ldquoPrevencionde la obesidad en los ninos desde el nacimiento hasta loscincoanosrdquo in Enciclopedia Sobre el Desarrollo de la PrimeraInfancia R Tremblay R Barr R Peters and M Boivin Edsvol 1ndash7 Centre of Excellence for EarlyChildhoodDevelopmentQuebec Canada 2010
[3] S K Garg H Maurer K Reed and R Selagamsetty ldquoDiabetesand cancer two diseases with obesity as a common risk factorrdquoDiabetes Obesity andMetabolism vol 16 no 2 pp 97ndash110 2014
[4] National Heart ldquoLung and Blood Institute World Health WhatAre the Health Risks of Overweight and Obesityrdquo httpswwwnhlbinihgovhealthhealth-topicstopicsoberisks
[5] World Health Organization (WHO) Global Health Observa-tory Data Repository WHO Geneva Switzerland 2011 httpappswhointghodata
[6] World Health Organization The Global Strategy on Diet Phys-ical Activity and Health (DPAS) WHO Geneva Switzerland2004 httpwwwwhointnmhwha59dpasen
[7] N A Christakis and J H Fowler ldquoThe spread of obesity in alarge social network over 32 yearsrdquoThe New England Journal ofMedicine vol 357 no 4 pp 370ndash379 2007
[8] K Ejima K Aihara and H Nishiura ldquoModeling the obesityepidemic social contagion and its implications for controlrdquoTheoretical Biology and Medical Modelling vol 10 article 17 pp1ndash13 2013
[9] A J Arenas G Gonzalez-Parra and L Jodar ldquoPeriodic solu-tions of nonautonomous differential systems modeling obesitypopulationrdquo Chaos Solitons amp Fractals vol 42 no 2 pp 1234ndash1244 2009
[10] G Gonzalez-Parra L Acedo R-J Villanueva Mico and A JArenas ldquoModeling the social obesity epidemic with stochasticnetworksrdquo Physica A Statistical Mechanics and its Applicationsvol 389 no 17 pp 3692ndash3701 2010
[11] G Gonzalez-Parra L Jodar F J Santonja and R J VillanuevaldquoAn age-structured model for childhood obesity mathematicalpopulation studiesrdquo An International Journal of MathematicalDemography vol 17 no 1 pp 1ndash11 2010
[12] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 Article ID e1000968 2010
[13] L Jodar F J Santonja and G Gonzalez-Parra ldquoModelingdynamics of infant obesity in the region of Valencia SpainrdquoComputers andMathematics with Applications vol 56 no 3 pp679ndash689 2008
[14] F-J Santonja R-J Villanueva L Jodar and G Gonzalez-Parra ldquoMathematical modelling of social obesity epidemic inthe region of Valencia Spainrdquo Mathematical and ComputerModelling of Dynamical Systems vol 16 no 1 pp 23ndash34 2010
[15] F J Santonja and L Shaikhet ldquoProbabilistic stability analysisof social obesity epidemic by a delayed stochastic modelrdquoNonlinear Analysis Real World Applications vol 17 pp 114ndash1252014
[16] D Tudor ldquoA deterministic model for herpes infections inhuman and animal populationsrdquo SIAM Review vol 32 no 1pp 136ndash139 1990
[17] A M Lyapunov The general problem of the stability of motionTaylor amp Francis London England 1992
[18] C Vargas-De-Leon ldquoLyapunov functions for two-species coop-erative systemsrdquo Applied Mathematics and Computation vol219 no 5 pp 2493ndash2497 2012
[19] D W Jordan and P Smith Nonlinear Ordinary DifferentialEquations An Introduction for Scientists and Engineers OxfordUniversity Press New York NY USA 4th edition 2007
[20] C M Kribs-Zaleta and J X Velasco-Hernandez ldquoA simplevaccination model with multiple endemic statesrdquoMathematicalBiosciences vol 164 no 2 pp 183ndash201 2000
[21] S Barquera I Campos-Nonato L Hernandez-Barrera APedroza-Tobıas and J A Rivera-Dommarco ldquoPrevalence ofobesity in mexican adults ensanut 2012rdquo Salud Publica deMexico vol 55 supplement 2 pp S151ndashS160 2013
the vector field of system (4) on the boundary point into thebox For the points on the horizontal side = 120574119868 gt 0 and = minus(119868 minus 119868+)(119868 minus 119868minus) where 119868+ and 119868minus are the positive andnegative roots of (120573119868 + 120598)(119868 minus 119873) + (120574 + 120583)119868 respectivelyThus these inequalities imply that the vector field points upand right on the interval (0 119868+) while it goes up and left on(119868+ 119873) Moreover in the vertical side = minus(120590120598 + 120583)119877 le 0and = 120598119873 + 120598(120590 minus 1)119877 gt 0 This implies that the vector fieldis directed downward and right except at the origin where itgoes to the right
Finally on the diagonal side of slope minus1 extending fromthe point (0119873) to (119873 0) it can be shown that
119889119877119889119868 = minus1 + 120583119873(119868 minus 119868lowast+) (119868 minus 119868lowast
minus) (37)
where 119868lowast+and 119868lowastminusare the positive and negative roots of 120590(120573119868 +120598)(119868 minus 119873) + (120574 + 120583)119868 respectively Hence the vector field
for large values of 119868 is almost parallel to the diagonal lineIn a more precise analysis always 119868 minus 119868lowast
minusgt 0 and 119868 minus 119868lowast
+gt0 if 119868 gt 119868lowast
+or 119868 minus 119868lowast
+lt 0 if 119868 lt 119868lowast
+ According to
(37) the first case implies that the vector field points inward(to the right) on the diagonal line because 119889119877119889119868 is morenegative than minus1 while the latter indicates that the vectorfield is directed inside (to the left) the diagonal side because119889119877119889119868 is less negative than minus1 (See Figure 4) Therefore weconclude that the triangular region is effectively a trappingregion
On the other hand Theorem 7 prohibits the existenceof closed orbits inside the triangular region Consequentlythe trajectories entering the triangular region accordingto the Poincare-Bendixson theorem [19] must converge tothe single equilibrium point located inside it Thereforethe equilibrium point inside the triangular region must beasymptotically stable Moreover since this point is the onlypoint of equilibrium in Δ it is also globally asymptoticallystable
The results of this section may be summarized in thefollowing theorem
Theorem 8 System (4) has a unique equilibrium point in thetriangular region Δ which is globally asymptotically stable
5 Numerical Results
In this section we perform a series of numerical simulationsand graphs of system (4) to illustrate the different resultsobtained for each of the two cases of interest 120598 gt 0 and120598 = 0 previously analyzed We will use the parameter valuesreported in the work developed by Ejima et al [8] theirepidemiological meanings and magnitudes are indicated inTable 1
51 Case 120598 = 0 According to the magnitudes of theparameters reported in Table 1 (19) takes the value 119877lowast
0=069363 whereas (21) acquires the value 119868lowast
0= 163304
Table 1 Baseline values and epidemiological meanings of theparameters given by Ejima et al [8]
Parameter Baseline values Epidemiological meaning
119873 100000 individuals(assumed) Population size
1120583 694 (per year) Average life expectancy at birth120573 296 times 10minus7 (peryear)
Transmission rate of obesity dueto social contagion120598 0012 (per year) Nonsocial contagion risk ofobesity120590 80 Relative hazard of obesity amongthe ex-obese1120574 358 (per year) Average duration of obesity
2000
20000
0
4000
4000
6000
600080
00
8000
1000
0
10000
1200
0
12000
1400
0
14000
1600
0
16000
1800
0
18000
2000
0
200002200024000260002800030000
2200
024
000
2600
028
000
3000
0
I (obese)
R (e
x-ob
ese)
Figure 5 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 03 obtained by considering 120573 = 127 times 10minus7Furthermore taking into consideration (10) and (11) theresulting two endemic equilibria are119864lowast
1= (20918 91378) (38)
119864lowast2= (12522 79384) (39)
which as discussed previously are asymptotically stable andunstable respectively
In Figures 5ndash9 we illustrated representative phase por-traits of the dynamic system (6) for some different values of1198770 between 0 and just over 1 according to the bifurcationdiagram shown in Figure 2 These values are obtained bymaintaining fixed in (9) 119873 120583 and 120574 but varying theparameter 120573 Thus if 1198770 takes values between 0 and 119877lowast
0
say 1198770 = 03 we have only one equilibrium point (thedisease-free equilibrium) which is globally asymptoticallystable (see Figure 5) If 1198770 = 119877lowast
0 we have two equilibrium
points one asymptotically stable disease-free equilibriumand the other endemic equilibria point which is nonhyper-bolic (see Figure 6) When 1198770 takes values between 119877lowast0 and
8 Discrete Dynamics in Nature and Society
2000
2000
0
0
4000
4000
6000
6000
8000
800010
000
10000
1200
0
12000
1400
0
14000
1600
0
16000
1800
0
18000
2000
0
20000
2200
0
2400
0I (obese)
R (e
x-ob
ese)
Figure 6 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 119877lowast0 obtained by considering 120573 = 294 times 10minus7
5000
2000
0
0
1000
0
4000
1500
0
6000
2000
0
8000
2500
0
10000
3000
0
12000
3500
0
14000
4000
0
16000
4500
0
18000
5000
0
20000
I (obese)
R (e
x-ob
ese)
Figure 7 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 08 obtained by considering 120573 = 339 times 10minus71 say 1198770 = 08 we have three equilibria one disease-free equilibrium (asymptotically stable) and two endemicequilibria (asymptotically stable and unstable) as shown inFigure 7 If 1198770 = 1 there are two equilibrium pointsone disease-free equilibrium which is nonhyperbolic and theother asymptotically stable endemic equilibrium as shown inFigure 8 Finally if 1198770 takes a slightly greater value to 1 forexample1198770 = 12 we have again two equilibrium points onedisease-free equilibrium (unstable) and the other endemicequilibrium (asymptotically stable) but both are hyperbolic(see Figure 9)
On the other hand in Figures 10ndash14 examples of theevolution in time of 119868(119905) and119877(119905) are presented for the values1198770 = 03 1198770 = 119877lowast
0 1198770 = 08 1198770 = 1 and 1198770 = 12 which
1000
00
2000
0
3000
0
4000
0
5000
0
6000
0
7000
0
8000
0
9000
0
1000
00
1100
00
1200
00
I (obese)
10000
20003000400050006000700080009000
10000110001200013000140001500016000
R (e
x-ob
ese)
Figure 8 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 1 obtained by considering 120573 = 423 times 10minus7
1000
0
10000
0
2000
0
2000
3000
0
30004000
4000
050
000
50006000
6000
07000
7000
0
80009000
8000
090
000
1000011000
1000
00
12000
1100
00
1400013000
1200
0013
0000
1400
00
1500016000
I (obese)
R (e
x-ob
ese)
Figure 9 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 12 obtained by considering 120573 = 508 times 10minus7illustrate respectively some of the dynamics shown in thephase portraits in Figures 5ndash9
52 Case 120598 gt 0 Taking into account the value 120598 =0012 gt 0 given in Table 1 in addition to the othersthere reported nullclines (33) and (34) intersect at the onlyendemic equilibrium point119864lowast = (60890 66805) (40)
which is globally asymptotically stable In Figure 15 thecorresponding phase portrait with several representativetrajectories around this equilibrium point is shown Also
Discrete Dynamics in Nature and Society 9
Time (years)
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 10 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 03 and the initial condition is(119868(0) 119877(0)) = (10000 30000)
Time (years)0 100 200 300
ObeseEx-obese
20000
15000
10000
5000
0
Figure 11 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 119877lowast
0and the initial condition is(119868(0) 119877(0)) = (20000 8000)
performing another series of numerical simulations of system(4) we show for this case in Figure 16 an example of theevolution in time of 119868(119905) and 119877(119905)6 Discussion and Conclusions
According to the results of our analysis of the Ejima et almodel we arrive at the following conclusions It has beenshown that the Ejima et al model [8] if only the socialcontagion (120598 = 0) is considered presents the two bifurcationvalues 1198770 = 119877lowast
0and 1198770 = 1 which results in what is
known as backward bifurcation From a mathematical pointof view the backward bifurcation is apparently constitutedby the combination of two different kinds of bifurcations
Time (years)
40000
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 12 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 08 and the initial condition is(119868(0) 119877(0)) = (30000 12000)
Time (years)
40000
50000
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 13 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 1 and the initial condition is(119868(0) 119877(0)) = (18000 2500)a saddle-node bifurcation that occurs at 1198770 = 119877lowast
0 whose
upper (asymptotically stable) and lower (unstable) quadraticbranches are defined for 119877lowast
0lt 1198770 and a transcritical
bifurcation that takes place in 1198770 = 1 whose horizontalbranches on the 1198770 axis are asymptotically stable for 1198770 lt 1and unstable for 1 lt 1198770 For values of 1 lt 1198770 only theupper quadratic branch asymptotically stable of the saddle-node bifurcation and the unstable horizontal branch of thetranscritical bifurcation have ameaning in the feasible regionΔ The diagram shown in Figure 2 suggests the presenceof these two bifurcations in the vicinities of the points(119877lowast0 119868lowast0) and (1 0) Of course we should go beyond the
graphic evidence and employ the qualitative analysis of localbifurcations in those points to verify our statement even
10 Discrete Dynamics in Nature and Society
Time (years)0 100 200 300
ObeseEx-obese
140000
120000
100000
80000
60000
40000
20000
0
Figure 14 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 12 and the initial condition is(119868(0) 119877(0)) = (100000 80000)
1000
0
10000
0
2000
0
2000
3000
0
30004000
4000
0
5000
0
50006000
6000
0
7000
7000
0
80009000
8000
0
9000
0
1000011000
1000
00
12000
1100
00
1400013000
1200
00
1300
00
1400
00
1500016000
I (obese)
R (e
x-ob
ese)
Figure 15 Phase portrait for system (4) for the case 120598 = 0012 gt 0(considering 120573 = 296 times 10minus7)without the realization of this depth analysis our discussionsand conclusions can be supported firmly from the resultsshown in this work of investigation
Indeed the presence of the different branches of familiesof equilibrium points shown in Figure 2 causes the system todisplay as the parameter1198770 is varied a bistable region and asa consequence the hysteresis phenomenon In order to showin more detail the presence of hysteresis in the bifurcationdiagram mentioned before suppose we start the system inthe state119860 (see Figure 17) and then slowly increase 1198770 (whichis indicated by an arrow to the right just beneath the axis1198770 of Figure 17) We remain at the origin until 1198770 = 1when the origin loses its stability At this point at the slightestdisturbance the system will jump to the state 119861 located in the
0
20000
40000
60000
80000
100000
120000
Time (years)0 100 200 300
ObeseEx-obese
Figure 16 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0012 gt 0 and the initial condition is(100000 50000)
B
A
1
C
80000
70000
60000
50000
40000
30000
20000
10000
0
Infe
cted
(I)
Rlowast0
Basic reproductive number (R0)
Figure 17 Schematic representation of hysteresis phenomenon thatexhibits the bifurcation diagram of system (4) for the case 120598 = 0Here the forward and backward paths are not equal as the parameter1198770 is changedstable upper branch Insofar as1198770 increases its value (1198770 gt 1)the equilibrium points of the system will move along thisbranch away from the state119861 If now1198770 continuously reducedits value equilibrium states will return to the state 119861 cross itand reach the state 119862 (in which 1198770 = 119877lowast
0) At this point at
the slightest change again equilibrium states will jump backto the origin (in 1198770 = 119877lowast0 ) and if 1198770 continues to decrease wewill return to the state119860 Consequently the system exhibits alack of reversibility or it is said that it has memory (or that itpresents hysteresis) because the forward and backward pathsare not identical
From an epidemiological point of view in this backwardbifurcation if 0 lt 1198770 lt 119877lowast
0 no matter where the initial
Discrete Dynamics in Nature and Society 11
conditions are taken solutions always tend to the straightline of stable disease-free points (that correspond to a never-obese population) When 119877lowast
0lt 1198770 lt 1 if the initial
conditions are located below the unstable quadratic branchof endemic equilibrium points (obese people) then solutionscontinue going to the horizontal line of stable disease-freepoints otherwise they will be directed to the stable curveof endemic points located at the upper quadratic branchFinally if 1 lt 1198770 no matter again where the initialconditions are situated solutions will continue heading forthe stable endemic points of such upper branch Should benoted that unlike what happens in systems that exhibit thephenomenon of forward bifurcation in which the endemicequilibrium exists only for 1198770 gt 1 [20] in systems exhibitinga backward bifurcation under certain initial conditions itis possible to have endemic states when 1198770 lt 1 In ourcase the endemic equilibrium that exists for 1198770 just aboveone has a large obese population so the result of 1198770 risingabove one would be a sudden and dramatic jump in thenumber of infective Moreover reducing 1198770 back below onewould not eradicate the obesity if the size of the infectedpopulation is greater or equal than 119868lowast
0 the corresponding
value of 119877lowast0 In order to eradicate the disease 1198770 must be
further reduced so it just needs to be slightly less than119877lowast0 to suddenly enter the region where only stable disease-
free equilibrium (never-obese population) exists Both suchdifferent abrupt behaviours are the result of hysteresisphenomenon
It can be seen from the bifurcation diagram shown inFigure 2 that theway inwhich obesity infected population canbe lowered (or raised) is through the reduction (or increase)of 1198770 parameter which in accordance with (9) depends onthe four quantities 119873 120573 120583 and 120574 Since we have assumedin (1) that the natural death and birth rates are equal anddenoted by 120583 the population size119873 is constantTherefore1198770can only change according to the following forms Firstly 1198770diminishes if the natural rate 120583 (of death or birth) increasesthe transmission rate 120573 decreases or the rate 120574 grows atwhich the infected individuals become recovered On theother hand 1198770 increment its value if the opposite occurs 120583decreases 120573 goes up or 120574 declines It should be noted thatchanges in 120583 could be slower and achieved in the long termprobably in decades which is not a viable option Converselychanges in 120573 or 120574 might be faster and perhaps occur infewer years Since the focus is on reducing obesity levels andtherefore the values of 1198770 it is of interest to seek mechanismsto decrease120573 andor increase 120574The former could be achievedby sensitizing at the population in general about the benefitsassociatedwith having a healthy and balanced diet and also bydesigning effective health and informative public preventionpoliciesThe latter could be done by providing better medicalassistance and making it available to the obese populationIt should be emphasized that this last strategy is not feasiblein countries with high levels of obesity as Mexico because atthe present time it is very expensive and will become evenmore expensive as the affected population increases In thisregard we believe that it is much more convenient that thegovernments of the affected countries implement broaderand more effective preventive policies
It is worth mentioning that in Mexico there have beenstudies to quantify more precisely the prevalence of peoplesuffering from overweight and obesity It is estimated that713 of Mexican adults aged 20 or more are found inthis condition [21] While trends show slowing down of theincrease in the obesity prevalence there is no evidence toinfer that prevalence will decrease in the coming years Inthis sense studies like this allow us to analyze the trends ofthe population with obesity and study the different possiblescenarios Particularly with the model studied this couldbe done for Mexico if we knew accurately the values ofits parameters 120573 and 120574 To our knowledge there is notsuch information Nevertheless these parameters could beestimated in a short or medium term based on statisti-cal information collected on obesity in our country Theknowledge of how these parameters behave over time wouldcontribute in quantifying the effectiveness of the Mexicanpublic policies against obesity
With respect to the model studied it is pertinent tocomment about the term of relapse 120590[120573119868(119905) + 120598]119877(119905) of thesecond equation (1) In the first place in this work it has beenconsidered that the parameter 120590 the relative risk of weightregain among ex-obese individuals is greater than unity (120590 gt1)This condition represents the high risk that recovered (ex-obese) individuals will become obese again However it isalso plausible to consider that 0 lt 120590 lt 1 which representsthat ex-obese population is more resistant to being obeseagain that is they are more aware of the health risks thatthis would represent The consideration that 0 lt 120590 lt 1 doesnot change the results of the stability analysis and periodicorbits performed in Sections 3 and 4 With respect to thelatter the nullcline 1198661 (given by (33)) changes to 1198661015840
1 where1198661015840
1= minus1198661 while 1198662 (34) remains the same The intersection
of the nullclines 11986610158401and 1198662 again occurs at only one point
in the region Δ and its stability is the same Regarding thisaspect we could conclude that regardless of the value ofthe parameter 120590 there is always the possibility that the ex-obese population is at risk of relapse and becoming obeseagain of course this possibility is less if 0 lt 120590 lt 1 andgreater if 120590 gt 1 In this sense public health programs againstobesity designed by governments may be more effectiveif they take into account the awareness of the ex-obesepopulation
Finally with regard to the mentioned relapse term120590[120573119868(119905) + 120598]119877(119905) if we consider only the effects of socialcontagion (which corresponds to 120598 = 0) it is reduced tothe nonlinear term 120590120573119868(119905)119877(119905) which causes system (4) toexhibit a backward bifurcation If the effects of nonsocialcontagion (that is 120598 gt 0) are also incorporated the presenceof the linear term 120590120573120598119877(119905) causes the backward bifurcationto disappear in system (4) and in its place it only presentsan equilibrium point global asymptotically stable The lattercase suggests that the way in which the effects of nonsocialcontagion are introduced into the model by means of thelinear part of the relapse term should be modified in orderto maintain much of the interesting dynamic that is presentin the backward bifurcation Certainly an alternative wayis considering instead of a linear a nonlinear contribution
12 Discrete Dynamics in Nature and Society
Perhaps it is possible to build amore general nonlinearmodelwhere in the case in which the contribution due to nonsocialeffects gradually fades the backward bifurcation of the Ejimaet al model could be obtained as a limit case
Competing Interests
The authors declare that they have no conflict of interests
Acknowledgments
This research was supported by the UACM-SECITI ResearchGrant PI-2013-34 Cruz Vargas-De-Leon would like toexpress his indebtedness to Dr Ramon Reyes Carreto for hishospitality during his stay as visiting professor at the UAGroin Chilpancingo
References
[1] World Health Organization Obesity Preventing and Managingthe Global Epidemic Report of a WHO Consultation WHOTechnical Report Series No 894 WHO Geneva Switzerland2000 httpswwwgooglecommxurlsa=tamprct=jampq=ampesrc=sampsource=webampcd=1ampved=0ahUKEwjlno 3tsDRAhVqi1QKH-W0rBPIQFggZMAAampurl=http3A2F2Fwhqlibdocwhoint2Ftrs2FWHO TRS 894pdfampusg=AFQjCNFLusNXrJz-xFKX1xpQwkHLEr-Lohgampbvm=bv144224172damcampcad=rja
[2] C L Vanvrancken-Tompkins andMaM Sothern ldquoPrevencionde la obesidad en los ninos desde el nacimiento hasta loscincoanosrdquo in Enciclopedia Sobre el Desarrollo de la PrimeraInfancia R Tremblay R Barr R Peters and M Boivin Edsvol 1ndash7 Centre of Excellence for EarlyChildhoodDevelopmentQuebec Canada 2010
[3] S K Garg H Maurer K Reed and R Selagamsetty ldquoDiabetesand cancer two diseases with obesity as a common risk factorrdquoDiabetes Obesity andMetabolism vol 16 no 2 pp 97ndash110 2014
[4] National Heart ldquoLung and Blood Institute World Health WhatAre the Health Risks of Overweight and Obesityrdquo httpswwwnhlbinihgovhealthhealth-topicstopicsoberisks
[5] World Health Organization (WHO) Global Health Observa-tory Data Repository WHO Geneva Switzerland 2011 httpappswhointghodata
[6] World Health Organization The Global Strategy on Diet Phys-ical Activity and Health (DPAS) WHO Geneva Switzerland2004 httpwwwwhointnmhwha59dpasen
[7] N A Christakis and J H Fowler ldquoThe spread of obesity in alarge social network over 32 yearsrdquoThe New England Journal ofMedicine vol 357 no 4 pp 370ndash379 2007
[8] K Ejima K Aihara and H Nishiura ldquoModeling the obesityepidemic social contagion and its implications for controlrdquoTheoretical Biology and Medical Modelling vol 10 article 17 pp1ndash13 2013
[9] A J Arenas G Gonzalez-Parra and L Jodar ldquoPeriodic solu-tions of nonautonomous differential systems modeling obesitypopulationrdquo Chaos Solitons amp Fractals vol 42 no 2 pp 1234ndash1244 2009
[10] G Gonzalez-Parra L Acedo R-J Villanueva Mico and A JArenas ldquoModeling the social obesity epidemic with stochasticnetworksrdquo Physica A Statistical Mechanics and its Applicationsvol 389 no 17 pp 3692ndash3701 2010
[11] G Gonzalez-Parra L Jodar F J Santonja and R J VillanuevaldquoAn age-structured model for childhood obesity mathematicalpopulation studiesrdquo An International Journal of MathematicalDemography vol 17 no 1 pp 1ndash11 2010
[12] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 Article ID e1000968 2010
[13] L Jodar F J Santonja and G Gonzalez-Parra ldquoModelingdynamics of infant obesity in the region of Valencia SpainrdquoComputers andMathematics with Applications vol 56 no 3 pp679ndash689 2008
[14] F-J Santonja R-J Villanueva L Jodar and G Gonzalez-Parra ldquoMathematical modelling of social obesity epidemic inthe region of Valencia Spainrdquo Mathematical and ComputerModelling of Dynamical Systems vol 16 no 1 pp 23ndash34 2010
[15] F J Santonja and L Shaikhet ldquoProbabilistic stability analysisof social obesity epidemic by a delayed stochastic modelrdquoNonlinear Analysis Real World Applications vol 17 pp 114ndash1252014
[16] D Tudor ldquoA deterministic model for herpes infections inhuman and animal populationsrdquo SIAM Review vol 32 no 1pp 136ndash139 1990
[17] A M Lyapunov The general problem of the stability of motionTaylor amp Francis London England 1992
[18] C Vargas-De-Leon ldquoLyapunov functions for two-species coop-erative systemsrdquo Applied Mathematics and Computation vol219 no 5 pp 2493ndash2497 2012
[19] D W Jordan and P Smith Nonlinear Ordinary DifferentialEquations An Introduction for Scientists and Engineers OxfordUniversity Press New York NY USA 4th edition 2007
[20] C M Kribs-Zaleta and J X Velasco-Hernandez ldquoA simplevaccination model with multiple endemic statesrdquoMathematicalBiosciences vol 164 no 2 pp 183ndash201 2000
[21] S Barquera I Campos-Nonato L Hernandez-Barrera APedroza-Tobıas and J A Rivera-Dommarco ldquoPrevalence ofobesity in mexican adults ensanut 2012rdquo Salud Publica deMexico vol 55 supplement 2 pp S151ndashS160 2013
Figure 6 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 119877lowast0 obtained by considering 120573 = 294 times 10minus7
5000
2000
0
0
1000
0
4000
1500
0
6000
2000
0
8000
2500
0
10000
3000
0
12000
3500
0
14000
4000
0
16000
4500
0
18000
5000
0
20000
I (obese)
R (e
x-ob
ese)
Figure 7 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 08 obtained by considering 120573 = 339 times 10minus71 say 1198770 = 08 we have three equilibria one disease-free equilibrium (asymptotically stable) and two endemicequilibria (asymptotically stable and unstable) as shown inFigure 7 If 1198770 = 1 there are two equilibrium pointsone disease-free equilibrium which is nonhyperbolic and theother asymptotically stable endemic equilibrium as shown inFigure 8 Finally if 1198770 takes a slightly greater value to 1 forexample1198770 = 12 we have again two equilibrium points onedisease-free equilibrium (unstable) and the other endemicequilibrium (asymptotically stable) but both are hyperbolic(see Figure 9)
On the other hand in Figures 10ndash14 examples of theevolution in time of 119868(119905) and119877(119905) are presented for the values1198770 = 03 1198770 = 119877lowast
0 1198770 = 08 1198770 = 1 and 1198770 = 12 which
1000
00
2000
0
3000
0
4000
0
5000
0
6000
0
7000
0
8000
0
9000
0
1000
00
1100
00
1200
00
I (obese)
10000
20003000400050006000700080009000
10000110001200013000140001500016000
R (e
x-ob
ese)
Figure 8 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 1 obtained by considering 120573 = 423 times 10minus7
1000
0
10000
0
2000
0
2000
3000
0
30004000
4000
050
000
50006000
6000
07000
7000
0
80009000
8000
090
000
1000011000
1000
00
12000
1100
00
1400013000
1200
0013
0000
1400
00
1500016000
I (obese)
R (e
x-ob
ese)
Figure 9 Phase portrait of system (6) (case 120598 = 0) for the value1198770 = 12 obtained by considering 120573 = 508 times 10minus7illustrate respectively some of the dynamics shown in thephase portraits in Figures 5ndash9
52 Case 120598 gt 0 Taking into account the value 120598 =0012 gt 0 given in Table 1 in addition to the othersthere reported nullclines (33) and (34) intersect at the onlyendemic equilibrium point119864lowast = (60890 66805) (40)
which is globally asymptotically stable In Figure 15 thecorresponding phase portrait with several representativetrajectories around this equilibrium point is shown Also
Discrete Dynamics in Nature and Society 9
Time (years)
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 10 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 03 and the initial condition is(119868(0) 119877(0)) = (10000 30000)
Time (years)0 100 200 300
ObeseEx-obese
20000
15000
10000
5000
0
Figure 11 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 119877lowast
0and the initial condition is(119868(0) 119877(0)) = (20000 8000)
performing another series of numerical simulations of system(4) we show for this case in Figure 16 an example of theevolution in time of 119868(119905) and 119877(119905)6 Discussion and Conclusions
According to the results of our analysis of the Ejima et almodel we arrive at the following conclusions It has beenshown that the Ejima et al model [8] if only the socialcontagion (120598 = 0) is considered presents the two bifurcationvalues 1198770 = 119877lowast
0and 1198770 = 1 which results in what is
known as backward bifurcation From a mathematical pointof view the backward bifurcation is apparently constitutedby the combination of two different kinds of bifurcations
Time (years)
40000
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 12 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 08 and the initial condition is(119868(0) 119877(0)) = (30000 12000)
Time (years)
40000
50000
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 13 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 1 and the initial condition is(119868(0) 119877(0)) = (18000 2500)a saddle-node bifurcation that occurs at 1198770 = 119877lowast
0 whose
upper (asymptotically stable) and lower (unstable) quadraticbranches are defined for 119877lowast
0lt 1198770 and a transcritical
bifurcation that takes place in 1198770 = 1 whose horizontalbranches on the 1198770 axis are asymptotically stable for 1198770 lt 1and unstable for 1 lt 1198770 For values of 1 lt 1198770 only theupper quadratic branch asymptotically stable of the saddle-node bifurcation and the unstable horizontal branch of thetranscritical bifurcation have ameaning in the feasible regionΔ The diagram shown in Figure 2 suggests the presenceof these two bifurcations in the vicinities of the points(119877lowast0 119868lowast0) and (1 0) Of course we should go beyond the
graphic evidence and employ the qualitative analysis of localbifurcations in those points to verify our statement even
10 Discrete Dynamics in Nature and Society
Time (years)0 100 200 300
ObeseEx-obese
140000
120000
100000
80000
60000
40000
20000
0
Figure 14 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 12 and the initial condition is(119868(0) 119877(0)) = (100000 80000)
1000
0
10000
0
2000
0
2000
3000
0
30004000
4000
0
5000
0
50006000
6000
0
7000
7000
0
80009000
8000
0
9000
0
1000011000
1000
00
12000
1100
00
1400013000
1200
00
1300
00
1400
00
1500016000
I (obese)
R (e
x-ob
ese)
Figure 15 Phase portrait for system (4) for the case 120598 = 0012 gt 0(considering 120573 = 296 times 10minus7)without the realization of this depth analysis our discussionsand conclusions can be supported firmly from the resultsshown in this work of investigation
Indeed the presence of the different branches of familiesof equilibrium points shown in Figure 2 causes the system todisplay as the parameter1198770 is varied a bistable region and asa consequence the hysteresis phenomenon In order to showin more detail the presence of hysteresis in the bifurcationdiagram mentioned before suppose we start the system inthe state119860 (see Figure 17) and then slowly increase 1198770 (whichis indicated by an arrow to the right just beneath the axis1198770 of Figure 17) We remain at the origin until 1198770 = 1when the origin loses its stability At this point at the slightestdisturbance the system will jump to the state 119861 located in the
0
20000
40000
60000
80000
100000
120000
Time (years)0 100 200 300
ObeseEx-obese
Figure 16 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0012 gt 0 and the initial condition is(100000 50000)
B
A
1
C
80000
70000
60000
50000
40000
30000
20000
10000
0
Infe
cted
(I)
Rlowast0
Basic reproductive number (R0)
Figure 17 Schematic representation of hysteresis phenomenon thatexhibits the bifurcation diagram of system (4) for the case 120598 = 0Here the forward and backward paths are not equal as the parameter1198770 is changedstable upper branch Insofar as1198770 increases its value (1198770 gt 1)the equilibrium points of the system will move along thisbranch away from the state119861 If now1198770 continuously reducedits value equilibrium states will return to the state 119861 cross itand reach the state 119862 (in which 1198770 = 119877lowast
0) At this point at
the slightest change again equilibrium states will jump backto the origin (in 1198770 = 119877lowast0 ) and if 1198770 continues to decrease wewill return to the state119860 Consequently the system exhibits alack of reversibility or it is said that it has memory (or that itpresents hysteresis) because the forward and backward pathsare not identical
From an epidemiological point of view in this backwardbifurcation if 0 lt 1198770 lt 119877lowast
0 no matter where the initial
Discrete Dynamics in Nature and Society 11
conditions are taken solutions always tend to the straightline of stable disease-free points (that correspond to a never-obese population) When 119877lowast
0lt 1198770 lt 1 if the initial
conditions are located below the unstable quadratic branchof endemic equilibrium points (obese people) then solutionscontinue going to the horizontal line of stable disease-freepoints otherwise they will be directed to the stable curveof endemic points located at the upper quadratic branchFinally if 1 lt 1198770 no matter again where the initialconditions are situated solutions will continue heading forthe stable endemic points of such upper branch Should benoted that unlike what happens in systems that exhibit thephenomenon of forward bifurcation in which the endemicequilibrium exists only for 1198770 gt 1 [20] in systems exhibitinga backward bifurcation under certain initial conditions itis possible to have endemic states when 1198770 lt 1 In ourcase the endemic equilibrium that exists for 1198770 just aboveone has a large obese population so the result of 1198770 risingabove one would be a sudden and dramatic jump in thenumber of infective Moreover reducing 1198770 back below onewould not eradicate the obesity if the size of the infectedpopulation is greater or equal than 119868lowast
0 the corresponding
value of 119877lowast0 In order to eradicate the disease 1198770 must be
further reduced so it just needs to be slightly less than119877lowast0 to suddenly enter the region where only stable disease-
free equilibrium (never-obese population) exists Both suchdifferent abrupt behaviours are the result of hysteresisphenomenon
It can be seen from the bifurcation diagram shown inFigure 2 that theway inwhich obesity infected population canbe lowered (or raised) is through the reduction (or increase)of 1198770 parameter which in accordance with (9) depends onthe four quantities 119873 120573 120583 and 120574 Since we have assumedin (1) that the natural death and birth rates are equal anddenoted by 120583 the population size119873 is constantTherefore1198770can only change according to the following forms Firstly 1198770diminishes if the natural rate 120583 (of death or birth) increasesthe transmission rate 120573 decreases or the rate 120574 grows atwhich the infected individuals become recovered On theother hand 1198770 increment its value if the opposite occurs 120583decreases 120573 goes up or 120574 declines It should be noted thatchanges in 120583 could be slower and achieved in the long termprobably in decades which is not a viable option Converselychanges in 120573 or 120574 might be faster and perhaps occur infewer years Since the focus is on reducing obesity levels andtherefore the values of 1198770 it is of interest to seek mechanismsto decrease120573 andor increase 120574The former could be achievedby sensitizing at the population in general about the benefitsassociatedwith having a healthy and balanced diet and also bydesigning effective health and informative public preventionpoliciesThe latter could be done by providing better medicalassistance and making it available to the obese populationIt should be emphasized that this last strategy is not feasiblein countries with high levels of obesity as Mexico because atthe present time it is very expensive and will become evenmore expensive as the affected population increases In thisregard we believe that it is much more convenient that thegovernments of the affected countries implement broaderand more effective preventive policies
It is worth mentioning that in Mexico there have beenstudies to quantify more precisely the prevalence of peoplesuffering from overweight and obesity It is estimated that713 of Mexican adults aged 20 or more are found inthis condition [21] While trends show slowing down of theincrease in the obesity prevalence there is no evidence toinfer that prevalence will decrease in the coming years Inthis sense studies like this allow us to analyze the trends ofthe population with obesity and study the different possiblescenarios Particularly with the model studied this couldbe done for Mexico if we knew accurately the values ofits parameters 120573 and 120574 To our knowledge there is notsuch information Nevertheless these parameters could beestimated in a short or medium term based on statisti-cal information collected on obesity in our country Theknowledge of how these parameters behave over time wouldcontribute in quantifying the effectiveness of the Mexicanpublic policies against obesity
With respect to the model studied it is pertinent tocomment about the term of relapse 120590[120573119868(119905) + 120598]119877(119905) of thesecond equation (1) In the first place in this work it has beenconsidered that the parameter 120590 the relative risk of weightregain among ex-obese individuals is greater than unity (120590 gt1)This condition represents the high risk that recovered (ex-obese) individuals will become obese again However it isalso plausible to consider that 0 lt 120590 lt 1 which representsthat ex-obese population is more resistant to being obeseagain that is they are more aware of the health risks thatthis would represent The consideration that 0 lt 120590 lt 1 doesnot change the results of the stability analysis and periodicorbits performed in Sections 3 and 4 With respect to thelatter the nullcline 1198661 (given by (33)) changes to 1198661015840
1 where1198661015840
1= minus1198661 while 1198662 (34) remains the same The intersection
of the nullclines 11986610158401and 1198662 again occurs at only one point
in the region Δ and its stability is the same Regarding thisaspect we could conclude that regardless of the value ofthe parameter 120590 there is always the possibility that the ex-obese population is at risk of relapse and becoming obeseagain of course this possibility is less if 0 lt 120590 lt 1 andgreater if 120590 gt 1 In this sense public health programs againstobesity designed by governments may be more effectiveif they take into account the awareness of the ex-obesepopulation
Finally with regard to the mentioned relapse term120590[120573119868(119905) + 120598]119877(119905) if we consider only the effects of socialcontagion (which corresponds to 120598 = 0) it is reduced tothe nonlinear term 120590120573119868(119905)119877(119905) which causes system (4) toexhibit a backward bifurcation If the effects of nonsocialcontagion (that is 120598 gt 0) are also incorporated the presenceof the linear term 120590120573120598119877(119905) causes the backward bifurcationto disappear in system (4) and in its place it only presentsan equilibrium point global asymptotically stable The lattercase suggests that the way in which the effects of nonsocialcontagion are introduced into the model by means of thelinear part of the relapse term should be modified in orderto maintain much of the interesting dynamic that is presentin the backward bifurcation Certainly an alternative wayis considering instead of a linear a nonlinear contribution
12 Discrete Dynamics in Nature and Society
Perhaps it is possible to build amore general nonlinearmodelwhere in the case in which the contribution due to nonsocialeffects gradually fades the backward bifurcation of the Ejimaet al model could be obtained as a limit case
Competing Interests
The authors declare that they have no conflict of interests
Acknowledgments
This research was supported by the UACM-SECITI ResearchGrant PI-2013-34 Cruz Vargas-De-Leon would like toexpress his indebtedness to Dr Ramon Reyes Carreto for hishospitality during his stay as visiting professor at the UAGroin Chilpancingo
References
[1] World Health Organization Obesity Preventing and Managingthe Global Epidemic Report of a WHO Consultation WHOTechnical Report Series No 894 WHO Geneva Switzerland2000 httpswwwgooglecommxurlsa=tamprct=jampq=ampesrc=sampsource=webampcd=1ampved=0ahUKEwjlno 3tsDRAhVqi1QKH-W0rBPIQFggZMAAampurl=http3A2F2Fwhqlibdocwhoint2Ftrs2FWHO TRS 894pdfampusg=AFQjCNFLusNXrJz-xFKX1xpQwkHLEr-Lohgampbvm=bv144224172damcampcad=rja
[2] C L Vanvrancken-Tompkins andMaM Sothern ldquoPrevencionde la obesidad en los ninos desde el nacimiento hasta loscincoanosrdquo in Enciclopedia Sobre el Desarrollo de la PrimeraInfancia R Tremblay R Barr R Peters and M Boivin Edsvol 1ndash7 Centre of Excellence for EarlyChildhoodDevelopmentQuebec Canada 2010
[3] S K Garg H Maurer K Reed and R Selagamsetty ldquoDiabetesand cancer two diseases with obesity as a common risk factorrdquoDiabetes Obesity andMetabolism vol 16 no 2 pp 97ndash110 2014
[4] National Heart ldquoLung and Blood Institute World Health WhatAre the Health Risks of Overweight and Obesityrdquo httpswwwnhlbinihgovhealthhealth-topicstopicsoberisks
[5] World Health Organization (WHO) Global Health Observa-tory Data Repository WHO Geneva Switzerland 2011 httpappswhointghodata
[6] World Health Organization The Global Strategy on Diet Phys-ical Activity and Health (DPAS) WHO Geneva Switzerland2004 httpwwwwhointnmhwha59dpasen
[7] N A Christakis and J H Fowler ldquoThe spread of obesity in alarge social network over 32 yearsrdquoThe New England Journal ofMedicine vol 357 no 4 pp 370ndash379 2007
[8] K Ejima K Aihara and H Nishiura ldquoModeling the obesityepidemic social contagion and its implications for controlrdquoTheoretical Biology and Medical Modelling vol 10 article 17 pp1ndash13 2013
[9] A J Arenas G Gonzalez-Parra and L Jodar ldquoPeriodic solu-tions of nonautonomous differential systems modeling obesitypopulationrdquo Chaos Solitons amp Fractals vol 42 no 2 pp 1234ndash1244 2009
[10] G Gonzalez-Parra L Acedo R-J Villanueva Mico and A JArenas ldquoModeling the social obesity epidemic with stochasticnetworksrdquo Physica A Statistical Mechanics and its Applicationsvol 389 no 17 pp 3692ndash3701 2010
[11] G Gonzalez-Parra L Jodar F J Santonja and R J VillanuevaldquoAn age-structured model for childhood obesity mathematicalpopulation studiesrdquo An International Journal of MathematicalDemography vol 17 no 1 pp 1ndash11 2010
[12] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 Article ID e1000968 2010
[13] L Jodar F J Santonja and G Gonzalez-Parra ldquoModelingdynamics of infant obesity in the region of Valencia SpainrdquoComputers andMathematics with Applications vol 56 no 3 pp679ndash689 2008
[14] F-J Santonja R-J Villanueva L Jodar and G Gonzalez-Parra ldquoMathematical modelling of social obesity epidemic inthe region of Valencia Spainrdquo Mathematical and ComputerModelling of Dynamical Systems vol 16 no 1 pp 23ndash34 2010
[15] F J Santonja and L Shaikhet ldquoProbabilistic stability analysisof social obesity epidemic by a delayed stochastic modelrdquoNonlinear Analysis Real World Applications vol 17 pp 114ndash1252014
[16] D Tudor ldquoA deterministic model for herpes infections inhuman and animal populationsrdquo SIAM Review vol 32 no 1pp 136ndash139 1990
[17] A M Lyapunov The general problem of the stability of motionTaylor amp Francis London England 1992
[18] C Vargas-De-Leon ldquoLyapunov functions for two-species coop-erative systemsrdquo Applied Mathematics and Computation vol219 no 5 pp 2493ndash2497 2012
[19] D W Jordan and P Smith Nonlinear Ordinary DifferentialEquations An Introduction for Scientists and Engineers OxfordUniversity Press New York NY USA 4th edition 2007
[20] C M Kribs-Zaleta and J X Velasco-Hernandez ldquoA simplevaccination model with multiple endemic statesrdquoMathematicalBiosciences vol 164 no 2 pp 183ndash201 2000
[21] S Barquera I Campos-Nonato L Hernandez-Barrera APedroza-Tobıas and J A Rivera-Dommarco ldquoPrevalence ofobesity in mexican adults ensanut 2012rdquo Salud Publica deMexico vol 55 supplement 2 pp S151ndashS160 2013
Figure 10 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 03 and the initial condition is(119868(0) 119877(0)) = (10000 30000)
Time (years)0 100 200 300
ObeseEx-obese
20000
15000
10000
5000
0
Figure 11 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 119877lowast
0and the initial condition is(119868(0) 119877(0)) = (20000 8000)
performing another series of numerical simulations of system(4) we show for this case in Figure 16 an example of theevolution in time of 119868(119905) and 119877(119905)6 Discussion and Conclusions
According to the results of our analysis of the Ejima et almodel we arrive at the following conclusions It has beenshown that the Ejima et al model [8] if only the socialcontagion (120598 = 0) is considered presents the two bifurcationvalues 1198770 = 119877lowast
0and 1198770 = 1 which results in what is
known as backward bifurcation From a mathematical pointof view the backward bifurcation is apparently constitutedby the combination of two different kinds of bifurcations
Time (years)
40000
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 12 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 08 and the initial condition is(119868(0) 119877(0)) = (30000 12000)
Time (years)
40000
50000
30000
20000
10000
00 100 200 300
ObeseEx-obese
Figure 13 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 1 and the initial condition is(119868(0) 119877(0)) = (18000 2500)a saddle-node bifurcation that occurs at 1198770 = 119877lowast
0 whose
upper (asymptotically stable) and lower (unstable) quadraticbranches are defined for 119877lowast
0lt 1198770 and a transcritical
bifurcation that takes place in 1198770 = 1 whose horizontalbranches on the 1198770 axis are asymptotically stable for 1198770 lt 1and unstable for 1 lt 1198770 For values of 1 lt 1198770 only theupper quadratic branch asymptotically stable of the saddle-node bifurcation and the unstable horizontal branch of thetranscritical bifurcation have ameaning in the feasible regionΔ The diagram shown in Figure 2 suggests the presenceof these two bifurcations in the vicinities of the points(119877lowast0 119868lowast0) and (1 0) Of course we should go beyond the
graphic evidence and employ the qualitative analysis of localbifurcations in those points to verify our statement even
10 Discrete Dynamics in Nature and Society
Time (years)0 100 200 300
ObeseEx-obese
140000
120000
100000
80000
60000
40000
20000
0
Figure 14 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 12 and the initial condition is(119868(0) 119877(0)) = (100000 80000)
1000
0
10000
0
2000
0
2000
3000
0
30004000
4000
0
5000
0
50006000
6000
0
7000
7000
0
80009000
8000
0
9000
0
1000011000
1000
00
12000
1100
00
1400013000
1200
00
1300
00
1400
00
1500016000
I (obese)
R (e
x-ob
ese)
Figure 15 Phase portrait for system (4) for the case 120598 = 0012 gt 0(considering 120573 = 296 times 10minus7)without the realization of this depth analysis our discussionsand conclusions can be supported firmly from the resultsshown in this work of investigation
Indeed the presence of the different branches of familiesof equilibrium points shown in Figure 2 causes the system todisplay as the parameter1198770 is varied a bistable region and asa consequence the hysteresis phenomenon In order to showin more detail the presence of hysteresis in the bifurcationdiagram mentioned before suppose we start the system inthe state119860 (see Figure 17) and then slowly increase 1198770 (whichis indicated by an arrow to the right just beneath the axis1198770 of Figure 17) We remain at the origin until 1198770 = 1when the origin loses its stability At this point at the slightestdisturbance the system will jump to the state 119861 located in the
0
20000
40000
60000
80000
100000
120000
Time (years)0 100 200 300
ObeseEx-obese
Figure 16 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0012 gt 0 and the initial condition is(100000 50000)
B
A
1
C
80000
70000
60000
50000
40000
30000
20000
10000
0
Infe
cted
(I)
Rlowast0
Basic reproductive number (R0)
Figure 17 Schematic representation of hysteresis phenomenon thatexhibits the bifurcation diagram of system (4) for the case 120598 = 0Here the forward and backward paths are not equal as the parameter1198770 is changedstable upper branch Insofar as1198770 increases its value (1198770 gt 1)the equilibrium points of the system will move along thisbranch away from the state119861 If now1198770 continuously reducedits value equilibrium states will return to the state 119861 cross itand reach the state 119862 (in which 1198770 = 119877lowast
0) At this point at
the slightest change again equilibrium states will jump backto the origin (in 1198770 = 119877lowast0 ) and if 1198770 continues to decrease wewill return to the state119860 Consequently the system exhibits alack of reversibility or it is said that it has memory (or that itpresents hysteresis) because the forward and backward pathsare not identical
From an epidemiological point of view in this backwardbifurcation if 0 lt 1198770 lt 119877lowast
0 no matter where the initial
Discrete Dynamics in Nature and Society 11
conditions are taken solutions always tend to the straightline of stable disease-free points (that correspond to a never-obese population) When 119877lowast
0lt 1198770 lt 1 if the initial
conditions are located below the unstable quadratic branchof endemic equilibrium points (obese people) then solutionscontinue going to the horizontal line of stable disease-freepoints otherwise they will be directed to the stable curveof endemic points located at the upper quadratic branchFinally if 1 lt 1198770 no matter again where the initialconditions are situated solutions will continue heading forthe stable endemic points of such upper branch Should benoted that unlike what happens in systems that exhibit thephenomenon of forward bifurcation in which the endemicequilibrium exists only for 1198770 gt 1 [20] in systems exhibitinga backward bifurcation under certain initial conditions itis possible to have endemic states when 1198770 lt 1 In ourcase the endemic equilibrium that exists for 1198770 just aboveone has a large obese population so the result of 1198770 risingabove one would be a sudden and dramatic jump in thenumber of infective Moreover reducing 1198770 back below onewould not eradicate the obesity if the size of the infectedpopulation is greater or equal than 119868lowast
0 the corresponding
value of 119877lowast0 In order to eradicate the disease 1198770 must be
further reduced so it just needs to be slightly less than119877lowast0 to suddenly enter the region where only stable disease-
free equilibrium (never-obese population) exists Both suchdifferent abrupt behaviours are the result of hysteresisphenomenon
It can be seen from the bifurcation diagram shown inFigure 2 that theway inwhich obesity infected population canbe lowered (or raised) is through the reduction (or increase)of 1198770 parameter which in accordance with (9) depends onthe four quantities 119873 120573 120583 and 120574 Since we have assumedin (1) that the natural death and birth rates are equal anddenoted by 120583 the population size119873 is constantTherefore1198770can only change according to the following forms Firstly 1198770diminishes if the natural rate 120583 (of death or birth) increasesthe transmission rate 120573 decreases or the rate 120574 grows atwhich the infected individuals become recovered On theother hand 1198770 increment its value if the opposite occurs 120583decreases 120573 goes up or 120574 declines It should be noted thatchanges in 120583 could be slower and achieved in the long termprobably in decades which is not a viable option Converselychanges in 120573 or 120574 might be faster and perhaps occur infewer years Since the focus is on reducing obesity levels andtherefore the values of 1198770 it is of interest to seek mechanismsto decrease120573 andor increase 120574The former could be achievedby sensitizing at the population in general about the benefitsassociatedwith having a healthy and balanced diet and also bydesigning effective health and informative public preventionpoliciesThe latter could be done by providing better medicalassistance and making it available to the obese populationIt should be emphasized that this last strategy is not feasiblein countries with high levels of obesity as Mexico because atthe present time it is very expensive and will become evenmore expensive as the affected population increases In thisregard we believe that it is much more convenient that thegovernments of the affected countries implement broaderand more effective preventive policies
It is worth mentioning that in Mexico there have beenstudies to quantify more precisely the prevalence of peoplesuffering from overweight and obesity It is estimated that713 of Mexican adults aged 20 or more are found inthis condition [21] While trends show slowing down of theincrease in the obesity prevalence there is no evidence toinfer that prevalence will decrease in the coming years Inthis sense studies like this allow us to analyze the trends ofthe population with obesity and study the different possiblescenarios Particularly with the model studied this couldbe done for Mexico if we knew accurately the values ofits parameters 120573 and 120574 To our knowledge there is notsuch information Nevertheless these parameters could beestimated in a short or medium term based on statisti-cal information collected on obesity in our country Theknowledge of how these parameters behave over time wouldcontribute in quantifying the effectiveness of the Mexicanpublic policies against obesity
With respect to the model studied it is pertinent tocomment about the term of relapse 120590[120573119868(119905) + 120598]119877(119905) of thesecond equation (1) In the first place in this work it has beenconsidered that the parameter 120590 the relative risk of weightregain among ex-obese individuals is greater than unity (120590 gt1)This condition represents the high risk that recovered (ex-obese) individuals will become obese again However it isalso plausible to consider that 0 lt 120590 lt 1 which representsthat ex-obese population is more resistant to being obeseagain that is they are more aware of the health risks thatthis would represent The consideration that 0 lt 120590 lt 1 doesnot change the results of the stability analysis and periodicorbits performed in Sections 3 and 4 With respect to thelatter the nullcline 1198661 (given by (33)) changes to 1198661015840
1 where1198661015840
1= minus1198661 while 1198662 (34) remains the same The intersection
of the nullclines 11986610158401and 1198662 again occurs at only one point
in the region Δ and its stability is the same Regarding thisaspect we could conclude that regardless of the value ofthe parameter 120590 there is always the possibility that the ex-obese population is at risk of relapse and becoming obeseagain of course this possibility is less if 0 lt 120590 lt 1 andgreater if 120590 gt 1 In this sense public health programs againstobesity designed by governments may be more effectiveif they take into account the awareness of the ex-obesepopulation
Finally with regard to the mentioned relapse term120590[120573119868(119905) + 120598]119877(119905) if we consider only the effects of socialcontagion (which corresponds to 120598 = 0) it is reduced tothe nonlinear term 120590120573119868(119905)119877(119905) which causes system (4) toexhibit a backward bifurcation If the effects of nonsocialcontagion (that is 120598 gt 0) are also incorporated the presenceof the linear term 120590120573120598119877(119905) causes the backward bifurcationto disappear in system (4) and in its place it only presentsan equilibrium point global asymptotically stable The lattercase suggests that the way in which the effects of nonsocialcontagion are introduced into the model by means of thelinear part of the relapse term should be modified in orderto maintain much of the interesting dynamic that is presentin the backward bifurcation Certainly an alternative wayis considering instead of a linear a nonlinear contribution
12 Discrete Dynamics in Nature and Society
Perhaps it is possible to build amore general nonlinearmodelwhere in the case in which the contribution due to nonsocialeffects gradually fades the backward bifurcation of the Ejimaet al model could be obtained as a limit case
Competing Interests
The authors declare that they have no conflict of interests
Acknowledgments
This research was supported by the UACM-SECITI ResearchGrant PI-2013-34 Cruz Vargas-De-Leon would like toexpress his indebtedness to Dr Ramon Reyes Carreto for hishospitality during his stay as visiting professor at the UAGroin Chilpancingo
References
[1] World Health Organization Obesity Preventing and Managingthe Global Epidemic Report of a WHO Consultation WHOTechnical Report Series No 894 WHO Geneva Switzerland2000 httpswwwgooglecommxurlsa=tamprct=jampq=ampesrc=sampsource=webampcd=1ampved=0ahUKEwjlno 3tsDRAhVqi1QKH-W0rBPIQFggZMAAampurl=http3A2F2Fwhqlibdocwhoint2Ftrs2FWHO TRS 894pdfampusg=AFQjCNFLusNXrJz-xFKX1xpQwkHLEr-Lohgampbvm=bv144224172damcampcad=rja
[2] C L Vanvrancken-Tompkins andMaM Sothern ldquoPrevencionde la obesidad en los ninos desde el nacimiento hasta loscincoanosrdquo in Enciclopedia Sobre el Desarrollo de la PrimeraInfancia R Tremblay R Barr R Peters and M Boivin Edsvol 1ndash7 Centre of Excellence for EarlyChildhoodDevelopmentQuebec Canada 2010
[3] S K Garg H Maurer K Reed and R Selagamsetty ldquoDiabetesand cancer two diseases with obesity as a common risk factorrdquoDiabetes Obesity andMetabolism vol 16 no 2 pp 97ndash110 2014
[4] National Heart ldquoLung and Blood Institute World Health WhatAre the Health Risks of Overweight and Obesityrdquo httpswwwnhlbinihgovhealthhealth-topicstopicsoberisks
[5] World Health Organization (WHO) Global Health Observa-tory Data Repository WHO Geneva Switzerland 2011 httpappswhointghodata
[6] World Health Organization The Global Strategy on Diet Phys-ical Activity and Health (DPAS) WHO Geneva Switzerland2004 httpwwwwhointnmhwha59dpasen
[7] N A Christakis and J H Fowler ldquoThe spread of obesity in alarge social network over 32 yearsrdquoThe New England Journal ofMedicine vol 357 no 4 pp 370ndash379 2007
[8] K Ejima K Aihara and H Nishiura ldquoModeling the obesityepidemic social contagion and its implications for controlrdquoTheoretical Biology and Medical Modelling vol 10 article 17 pp1ndash13 2013
[9] A J Arenas G Gonzalez-Parra and L Jodar ldquoPeriodic solu-tions of nonautonomous differential systems modeling obesitypopulationrdquo Chaos Solitons amp Fractals vol 42 no 2 pp 1234ndash1244 2009
[10] G Gonzalez-Parra L Acedo R-J Villanueva Mico and A JArenas ldquoModeling the social obesity epidemic with stochasticnetworksrdquo Physica A Statistical Mechanics and its Applicationsvol 389 no 17 pp 3692ndash3701 2010
[11] G Gonzalez-Parra L Jodar F J Santonja and R J VillanuevaldquoAn age-structured model for childhood obesity mathematicalpopulation studiesrdquo An International Journal of MathematicalDemography vol 17 no 1 pp 1ndash11 2010
[12] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 Article ID e1000968 2010
[13] L Jodar F J Santonja and G Gonzalez-Parra ldquoModelingdynamics of infant obesity in the region of Valencia SpainrdquoComputers andMathematics with Applications vol 56 no 3 pp679ndash689 2008
[14] F-J Santonja R-J Villanueva L Jodar and G Gonzalez-Parra ldquoMathematical modelling of social obesity epidemic inthe region of Valencia Spainrdquo Mathematical and ComputerModelling of Dynamical Systems vol 16 no 1 pp 23ndash34 2010
[15] F J Santonja and L Shaikhet ldquoProbabilistic stability analysisof social obesity epidemic by a delayed stochastic modelrdquoNonlinear Analysis Real World Applications vol 17 pp 114ndash1252014
[16] D Tudor ldquoA deterministic model for herpes infections inhuman and animal populationsrdquo SIAM Review vol 32 no 1pp 136ndash139 1990
[17] A M Lyapunov The general problem of the stability of motionTaylor amp Francis London England 1992
[18] C Vargas-De-Leon ldquoLyapunov functions for two-species coop-erative systemsrdquo Applied Mathematics and Computation vol219 no 5 pp 2493ndash2497 2012
[19] D W Jordan and P Smith Nonlinear Ordinary DifferentialEquations An Introduction for Scientists and Engineers OxfordUniversity Press New York NY USA 4th edition 2007
[20] C M Kribs-Zaleta and J X Velasco-Hernandez ldquoA simplevaccination model with multiple endemic statesrdquoMathematicalBiosciences vol 164 no 2 pp 183ndash201 2000
[21] S Barquera I Campos-Nonato L Hernandez-Barrera APedroza-Tobıas and J A Rivera-Dommarco ldquoPrevalence ofobesity in mexican adults ensanut 2012rdquo Salud Publica deMexico vol 55 supplement 2 pp S151ndashS160 2013
Figure 14 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0 1198770 = 12 and the initial condition is(119868(0) 119877(0)) = (100000 80000)
1000
0
10000
0
2000
0
2000
3000
0
30004000
4000
0
5000
0
50006000
6000
0
7000
7000
0
80009000
8000
0
9000
0
1000011000
1000
00
12000
1100
00
1400013000
1200
00
1300
00
1400
00
1500016000
I (obese)
R (e
x-ob
ese)
Figure 15 Phase portrait for system (4) for the case 120598 = 0012 gt 0(considering 120573 = 296 times 10minus7)without the realization of this depth analysis our discussionsand conclusions can be supported firmly from the resultsshown in this work of investigation
Indeed the presence of the different branches of familiesof equilibrium points shown in Figure 2 causes the system todisplay as the parameter1198770 is varied a bistable region and asa consequence the hysteresis phenomenon In order to showin more detail the presence of hysteresis in the bifurcationdiagram mentioned before suppose we start the system inthe state119860 (see Figure 17) and then slowly increase 1198770 (whichis indicated by an arrow to the right just beneath the axis1198770 of Figure 17) We remain at the origin until 1198770 = 1when the origin loses its stability At this point at the slightestdisturbance the system will jump to the state 119861 located in the
0
20000
40000
60000
80000
100000
120000
Time (years)0 100 200 300
ObeseEx-obese
Figure 16 Infected (obese) and recovered (ex-obese) individuals asfunction of time Here 120598 = 0012 gt 0 and the initial condition is(100000 50000)
B
A
1
C
80000
70000
60000
50000
40000
30000
20000
10000
0
Infe
cted
(I)
Rlowast0
Basic reproductive number (R0)
Figure 17 Schematic representation of hysteresis phenomenon thatexhibits the bifurcation diagram of system (4) for the case 120598 = 0Here the forward and backward paths are not equal as the parameter1198770 is changedstable upper branch Insofar as1198770 increases its value (1198770 gt 1)the equilibrium points of the system will move along thisbranch away from the state119861 If now1198770 continuously reducedits value equilibrium states will return to the state 119861 cross itand reach the state 119862 (in which 1198770 = 119877lowast
0) At this point at
the slightest change again equilibrium states will jump backto the origin (in 1198770 = 119877lowast0 ) and if 1198770 continues to decrease wewill return to the state119860 Consequently the system exhibits alack of reversibility or it is said that it has memory (or that itpresents hysteresis) because the forward and backward pathsare not identical
From an epidemiological point of view in this backwardbifurcation if 0 lt 1198770 lt 119877lowast
0 no matter where the initial
Discrete Dynamics in Nature and Society 11
conditions are taken solutions always tend to the straightline of stable disease-free points (that correspond to a never-obese population) When 119877lowast
0lt 1198770 lt 1 if the initial
conditions are located below the unstable quadratic branchof endemic equilibrium points (obese people) then solutionscontinue going to the horizontal line of stable disease-freepoints otherwise they will be directed to the stable curveof endemic points located at the upper quadratic branchFinally if 1 lt 1198770 no matter again where the initialconditions are situated solutions will continue heading forthe stable endemic points of such upper branch Should benoted that unlike what happens in systems that exhibit thephenomenon of forward bifurcation in which the endemicequilibrium exists only for 1198770 gt 1 [20] in systems exhibitinga backward bifurcation under certain initial conditions itis possible to have endemic states when 1198770 lt 1 In ourcase the endemic equilibrium that exists for 1198770 just aboveone has a large obese population so the result of 1198770 risingabove one would be a sudden and dramatic jump in thenumber of infective Moreover reducing 1198770 back below onewould not eradicate the obesity if the size of the infectedpopulation is greater or equal than 119868lowast
0 the corresponding
value of 119877lowast0 In order to eradicate the disease 1198770 must be
further reduced so it just needs to be slightly less than119877lowast0 to suddenly enter the region where only stable disease-
free equilibrium (never-obese population) exists Both suchdifferent abrupt behaviours are the result of hysteresisphenomenon
It can be seen from the bifurcation diagram shown inFigure 2 that theway inwhich obesity infected population canbe lowered (or raised) is through the reduction (or increase)of 1198770 parameter which in accordance with (9) depends onthe four quantities 119873 120573 120583 and 120574 Since we have assumedin (1) that the natural death and birth rates are equal anddenoted by 120583 the population size119873 is constantTherefore1198770can only change according to the following forms Firstly 1198770diminishes if the natural rate 120583 (of death or birth) increasesthe transmission rate 120573 decreases or the rate 120574 grows atwhich the infected individuals become recovered On theother hand 1198770 increment its value if the opposite occurs 120583decreases 120573 goes up or 120574 declines It should be noted thatchanges in 120583 could be slower and achieved in the long termprobably in decades which is not a viable option Converselychanges in 120573 or 120574 might be faster and perhaps occur infewer years Since the focus is on reducing obesity levels andtherefore the values of 1198770 it is of interest to seek mechanismsto decrease120573 andor increase 120574The former could be achievedby sensitizing at the population in general about the benefitsassociatedwith having a healthy and balanced diet and also bydesigning effective health and informative public preventionpoliciesThe latter could be done by providing better medicalassistance and making it available to the obese populationIt should be emphasized that this last strategy is not feasiblein countries with high levels of obesity as Mexico because atthe present time it is very expensive and will become evenmore expensive as the affected population increases In thisregard we believe that it is much more convenient that thegovernments of the affected countries implement broaderand more effective preventive policies
It is worth mentioning that in Mexico there have beenstudies to quantify more precisely the prevalence of peoplesuffering from overweight and obesity It is estimated that713 of Mexican adults aged 20 or more are found inthis condition [21] While trends show slowing down of theincrease in the obesity prevalence there is no evidence toinfer that prevalence will decrease in the coming years Inthis sense studies like this allow us to analyze the trends ofthe population with obesity and study the different possiblescenarios Particularly with the model studied this couldbe done for Mexico if we knew accurately the values ofits parameters 120573 and 120574 To our knowledge there is notsuch information Nevertheless these parameters could beestimated in a short or medium term based on statisti-cal information collected on obesity in our country Theknowledge of how these parameters behave over time wouldcontribute in quantifying the effectiveness of the Mexicanpublic policies against obesity
With respect to the model studied it is pertinent tocomment about the term of relapse 120590[120573119868(119905) + 120598]119877(119905) of thesecond equation (1) In the first place in this work it has beenconsidered that the parameter 120590 the relative risk of weightregain among ex-obese individuals is greater than unity (120590 gt1)This condition represents the high risk that recovered (ex-obese) individuals will become obese again However it isalso plausible to consider that 0 lt 120590 lt 1 which representsthat ex-obese population is more resistant to being obeseagain that is they are more aware of the health risks thatthis would represent The consideration that 0 lt 120590 lt 1 doesnot change the results of the stability analysis and periodicorbits performed in Sections 3 and 4 With respect to thelatter the nullcline 1198661 (given by (33)) changes to 1198661015840
1 where1198661015840
1= minus1198661 while 1198662 (34) remains the same The intersection
of the nullclines 11986610158401and 1198662 again occurs at only one point
in the region Δ and its stability is the same Regarding thisaspect we could conclude that regardless of the value ofthe parameter 120590 there is always the possibility that the ex-obese population is at risk of relapse and becoming obeseagain of course this possibility is less if 0 lt 120590 lt 1 andgreater if 120590 gt 1 In this sense public health programs againstobesity designed by governments may be more effectiveif they take into account the awareness of the ex-obesepopulation
Finally with regard to the mentioned relapse term120590[120573119868(119905) + 120598]119877(119905) if we consider only the effects of socialcontagion (which corresponds to 120598 = 0) it is reduced tothe nonlinear term 120590120573119868(119905)119877(119905) which causes system (4) toexhibit a backward bifurcation If the effects of nonsocialcontagion (that is 120598 gt 0) are also incorporated the presenceof the linear term 120590120573120598119877(119905) causes the backward bifurcationto disappear in system (4) and in its place it only presentsan equilibrium point global asymptotically stable The lattercase suggests that the way in which the effects of nonsocialcontagion are introduced into the model by means of thelinear part of the relapse term should be modified in orderto maintain much of the interesting dynamic that is presentin the backward bifurcation Certainly an alternative wayis considering instead of a linear a nonlinear contribution
12 Discrete Dynamics in Nature and Society
Perhaps it is possible to build amore general nonlinearmodelwhere in the case in which the contribution due to nonsocialeffects gradually fades the backward bifurcation of the Ejimaet al model could be obtained as a limit case
Competing Interests
The authors declare that they have no conflict of interests
Acknowledgments
This research was supported by the UACM-SECITI ResearchGrant PI-2013-34 Cruz Vargas-De-Leon would like toexpress his indebtedness to Dr Ramon Reyes Carreto for hishospitality during his stay as visiting professor at the UAGroin Chilpancingo
References
[1] World Health Organization Obesity Preventing and Managingthe Global Epidemic Report of a WHO Consultation WHOTechnical Report Series No 894 WHO Geneva Switzerland2000 httpswwwgooglecommxurlsa=tamprct=jampq=ampesrc=sampsource=webampcd=1ampved=0ahUKEwjlno 3tsDRAhVqi1QKH-W0rBPIQFggZMAAampurl=http3A2F2Fwhqlibdocwhoint2Ftrs2FWHO TRS 894pdfampusg=AFQjCNFLusNXrJz-xFKX1xpQwkHLEr-Lohgampbvm=bv144224172damcampcad=rja
[2] C L Vanvrancken-Tompkins andMaM Sothern ldquoPrevencionde la obesidad en los ninos desde el nacimiento hasta loscincoanosrdquo in Enciclopedia Sobre el Desarrollo de la PrimeraInfancia R Tremblay R Barr R Peters and M Boivin Edsvol 1ndash7 Centre of Excellence for EarlyChildhoodDevelopmentQuebec Canada 2010
[3] S K Garg H Maurer K Reed and R Selagamsetty ldquoDiabetesand cancer two diseases with obesity as a common risk factorrdquoDiabetes Obesity andMetabolism vol 16 no 2 pp 97ndash110 2014
[4] National Heart ldquoLung and Blood Institute World Health WhatAre the Health Risks of Overweight and Obesityrdquo httpswwwnhlbinihgovhealthhealth-topicstopicsoberisks
[5] World Health Organization (WHO) Global Health Observa-tory Data Repository WHO Geneva Switzerland 2011 httpappswhointghodata
[6] World Health Organization The Global Strategy on Diet Phys-ical Activity and Health (DPAS) WHO Geneva Switzerland2004 httpwwwwhointnmhwha59dpasen
[7] N A Christakis and J H Fowler ldquoThe spread of obesity in alarge social network over 32 yearsrdquoThe New England Journal ofMedicine vol 357 no 4 pp 370ndash379 2007
[8] K Ejima K Aihara and H Nishiura ldquoModeling the obesityepidemic social contagion and its implications for controlrdquoTheoretical Biology and Medical Modelling vol 10 article 17 pp1ndash13 2013
[9] A J Arenas G Gonzalez-Parra and L Jodar ldquoPeriodic solu-tions of nonautonomous differential systems modeling obesitypopulationrdquo Chaos Solitons amp Fractals vol 42 no 2 pp 1234ndash1244 2009
[10] G Gonzalez-Parra L Acedo R-J Villanueva Mico and A JArenas ldquoModeling the social obesity epidemic with stochasticnetworksrdquo Physica A Statistical Mechanics and its Applicationsvol 389 no 17 pp 3692ndash3701 2010
[11] G Gonzalez-Parra L Jodar F J Santonja and R J VillanuevaldquoAn age-structured model for childhood obesity mathematicalpopulation studiesrdquo An International Journal of MathematicalDemography vol 17 no 1 pp 1ndash11 2010
[12] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 Article ID e1000968 2010
[13] L Jodar F J Santonja and G Gonzalez-Parra ldquoModelingdynamics of infant obesity in the region of Valencia SpainrdquoComputers andMathematics with Applications vol 56 no 3 pp679ndash689 2008
[14] F-J Santonja R-J Villanueva L Jodar and G Gonzalez-Parra ldquoMathematical modelling of social obesity epidemic inthe region of Valencia Spainrdquo Mathematical and ComputerModelling of Dynamical Systems vol 16 no 1 pp 23ndash34 2010
[15] F J Santonja and L Shaikhet ldquoProbabilistic stability analysisof social obesity epidemic by a delayed stochastic modelrdquoNonlinear Analysis Real World Applications vol 17 pp 114ndash1252014
[16] D Tudor ldquoA deterministic model for herpes infections inhuman and animal populationsrdquo SIAM Review vol 32 no 1pp 136ndash139 1990
[17] A M Lyapunov The general problem of the stability of motionTaylor amp Francis London England 1992
[18] C Vargas-De-Leon ldquoLyapunov functions for two-species coop-erative systemsrdquo Applied Mathematics and Computation vol219 no 5 pp 2493ndash2497 2012
[19] D W Jordan and P Smith Nonlinear Ordinary DifferentialEquations An Introduction for Scientists and Engineers OxfordUniversity Press New York NY USA 4th edition 2007
[20] C M Kribs-Zaleta and J X Velasco-Hernandez ldquoA simplevaccination model with multiple endemic statesrdquoMathematicalBiosciences vol 164 no 2 pp 183ndash201 2000
[21] S Barquera I Campos-Nonato L Hernandez-Barrera APedroza-Tobıas and J A Rivera-Dommarco ldquoPrevalence ofobesity in mexican adults ensanut 2012rdquo Salud Publica deMexico vol 55 supplement 2 pp S151ndashS160 2013
conditions are taken solutions always tend to the straightline of stable disease-free points (that correspond to a never-obese population) When 119877lowast
0lt 1198770 lt 1 if the initial
conditions are located below the unstable quadratic branchof endemic equilibrium points (obese people) then solutionscontinue going to the horizontal line of stable disease-freepoints otherwise they will be directed to the stable curveof endemic points located at the upper quadratic branchFinally if 1 lt 1198770 no matter again where the initialconditions are situated solutions will continue heading forthe stable endemic points of such upper branch Should benoted that unlike what happens in systems that exhibit thephenomenon of forward bifurcation in which the endemicequilibrium exists only for 1198770 gt 1 [20] in systems exhibitinga backward bifurcation under certain initial conditions itis possible to have endemic states when 1198770 lt 1 In ourcase the endemic equilibrium that exists for 1198770 just aboveone has a large obese population so the result of 1198770 risingabove one would be a sudden and dramatic jump in thenumber of infective Moreover reducing 1198770 back below onewould not eradicate the obesity if the size of the infectedpopulation is greater or equal than 119868lowast
0 the corresponding
value of 119877lowast0 In order to eradicate the disease 1198770 must be
further reduced so it just needs to be slightly less than119877lowast0 to suddenly enter the region where only stable disease-
free equilibrium (never-obese population) exists Both suchdifferent abrupt behaviours are the result of hysteresisphenomenon
It can be seen from the bifurcation diagram shown inFigure 2 that theway inwhich obesity infected population canbe lowered (or raised) is through the reduction (or increase)of 1198770 parameter which in accordance with (9) depends onthe four quantities 119873 120573 120583 and 120574 Since we have assumedin (1) that the natural death and birth rates are equal anddenoted by 120583 the population size119873 is constantTherefore1198770can only change according to the following forms Firstly 1198770diminishes if the natural rate 120583 (of death or birth) increasesthe transmission rate 120573 decreases or the rate 120574 grows atwhich the infected individuals become recovered On theother hand 1198770 increment its value if the opposite occurs 120583decreases 120573 goes up or 120574 declines It should be noted thatchanges in 120583 could be slower and achieved in the long termprobably in decades which is not a viable option Converselychanges in 120573 or 120574 might be faster and perhaps occur infewer years Since the focus is on reducing obesity levels andtherefore the values of 1198770 it is of interest to seek mechanismsto decrease120573 andor increase 120574The former could be achievedby sensitizing at the population in general about the benefitsassociatedwith having a healthy and balanced diet and also bydesigning effective health and informative public preventionpoliciesThe latter could be done by providing better medicalassistance and making it available to the obese populationIt should be emphasized that this last strategy is not feasiblein countries with high levels of obesity as Mexico because atthe present time it is very expensive and will become evenmore expensive as the affected population increases In thisregard we believe that it is much more convenient that thegovernments of the affected countries implement broaderand more effective preventive policies
It is worth mentioning that in Mexico there have beenstudies to quantify more precisely the prevalence of peoplesuffering from overweight and obesity It is estimated that713 of Mexican adults aged 20 or more are found inthis condition [21] While trends show slowing down of theincrease in the obesity prevalence there is no evidence toinfer that prevalence will decrease in the coming years Inthis sense studies like this allow us to analyze the trends ofthe population with obesity and study the different possiblescenarios Particularly with the model studied this couldbe done for Mexico if we knew accurately the values ofits parameters 120573 and 120574 To our knowledge there is notsuch information Nevertheless these parameters could beestimated in a short or medium term based on statisti-cal information collected on obesity in our country Theknowledge of how these parameters behave over time wouldcontribute in quantifying the effectiveness of the Mexicanpublic policies against obesity
With respect to the model studied it is pertinent tocomment about the term of relapse 120590[120573119868(119905) + 120598]119877(119905) of thesecond equation (1) In the first place in this work it has beenconsidered that the parameter 120590 the relative risk of weightregain among ex-obese individuals is greater than unity (120590 gt1)This condition represents the high risk that recovered (ex-obese) individuals will become obese again However it isalso plausible to consider that 0 lt 120590 lt 1 which representsthat ex-obese population is more resistant to being obeseagain that is they are more aware of the health risks thatthis would represent The consideration that 0 lt 120590 lt 1 doesnot change the results of the stability analysis and periodicorbits performed in Sections 3 and 4 With respect to thelatter the nullcline 1198661 (given by (33)) changes to 1198661015840
1 where1198661015840
1= minus1198661 while 1198662 (34) remains the same The intersection
of the nullclines 11986610158401and 1198662 again occurs at only one point
in the region Δ and its stability is the same Regarding thisaspect we could conclude that regardless of the value ofthe parameter 120590 there is always the possibility that the ex-obese population is at risk of relapse and becoming obeseagain of course this possibility is less if 0 lt 120590 lt 1 andgreater if 120590 gt 1 In this sense public health programs againstobesity designed by governments may be more effectiveif they take into account the awareness of the ex-obesepopulation
Finally with regard to the mentioned relapse term120590[120573119868(119905) + 120598]119877(119905) if we consider only the effects of socialcontagion (which corresponds to 120598 = 0) it is reduced tothe nonlinear term 120590120573119868(119905)119877(119905) which causes system (4) toexhibit a backward bifurcation If the effects of nonsocialcontagion (that is 120598 gt 0) are also incorporated the presenceof the linear term 120590120573120598119877(119905) causes the backward bifurcationto disappear in system (4) and in its place it only presentsan equilibrium point global asymptotically stable The lattercase suggests that the way in which the effects of nonsocialcontagion are introduced into the model by means of thelinear part of the relapse term should be modified in orderto maintain much of the interesting dynamic that is presentin the backward bifurcation Certainly an alternative wayis considering instead of a linear a nonlinear contribution
12 Discrete Dynamics in Nature and Society
Perhaps it is possible to build amore general nonlinearmodelwhere in the case in which the contribution due to nonsocialeffects gradually fades the backward bifurcation of the Ejimaet al model could be obtained as a limit case
Competing Interests
The authors declare that they have no conflict of interests
Acknowledgments
This research was supported by the UACM-SECITI ResearchGrant PI-2013-34 Cruz Vargas-De-Leon would like toexpress his indebtedness to Dr Ramon Reyes Carreto for hishospitality during his stay as visiting professor at the UAGroin Chilpancingo
References
[1] World Health Organization Obesity Preventing and Managingthe Global Epidemic Report of a WHO Consultation WHOTechnical Report Series No 894 WHO Geneva Switzerland2000 httpswwwgooglecommxurlsa=tamprct=jampq=ampesrc=sampsource=webampcd=1ampved=0ahUKEwjlno 3tsDRAhVqi1QKH-W0rBPIQFggZMAAampurl=http3A2F2Fwhqlibdocwhoint2Ftrs2FWHO TRS 894pdfampusg=AFQjCNFLusNXrJz-xFKX1xpQwkHLEr-Lohgampbvm=bv144224172damcampcad=rja
[2] C L Vanvrancken-Tompkins andMaM Sothern ldquoPrevencionde la obesidad en los ninos desde el nacimiento hasta loscincoanosrdquo in Enciclopedia Sobre el Desarrollo de la PrimeraInfancia R Tremblay R Barr R Peters and M Boivin Edsvol 1ndash7 Centre of Excellence for EarlyChildhoodDevelopmentQuebec Canada 2010
[3] S K Garg H Maurer K Reed and R Selagamsetty ldquoDiabetesand cancer two diseases with obesity as a common risk factorrdquoDiabetes Obesity andMetabolism vol 16 no 2 pp 97ndash110 2014
[4] National Heart ldquoLung and Blood Institute World Health WhatAre the Health Risks of Overweight and Obesityrdquo httpswwwnhlbinihgovhealthhealth-topicstopicsoberisks
[5] World Health Organization (WHO) Global Health Observa-tory Data Repository WHO Geneva Switzerland 2011 httpappswhointghodata
[6] World Health Organization The Global Strategy on Diet Phys-ical Activity and Health (DPAS) WHO Geneva Switzerland2004 httpwwwwhointnmhwha59dpasen
[7] N A Christakis and J H Fowler ldquoThe spread of obesity in alarge social network over 32 yearsrdquoThe New England Journal ofMedicine vol 357 no 4 pp 370ndash379 2007
[8] K Ejima K Aihara and H Nishiura ldquoModeling the obesityepidemic social contagion and its implications for controlrdquoTheoretical Biology and Medical Modelling vol 10 article 17 pp1ndash13 2013
[9] A J Arenas G Gonzalez-Parra and L Jodar ldquoPeriodic solu-tions of nonautonomous differential systems modeling obesitypopulationrdquo Chaos Solitons amp Fractals vol 42 no 2 pp 1234ndash1244 2009
[10] G Gonzalez-Parra L Acedo R-J Villanueva Mico and A JArenas ldquoModeling the social obesity epidemic with stochasticnetworksrdquo Physica A Statistical Mechanics and its Applicationsvol 389 no 17 pp 3692ndash3701 2010
[11] G Gonzalez-Parra L Jodar F J Santonja and R J VillanuevaldquoAn age-structured model for childhood obesity mathematicalpopulation studiesrdquo An International Journal of MathematicalDemography vol 17 no 1 pp 1ndash11 2010
[12] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 Article ID e1000968 2010
[13] L Jodar F J Santonja and G Gonzalez-Parra ldquoModelingdynamics of infant obesity in the region of Valencia SpainrdquoComputers andMathematics with Applications vol 56 no 3 pp679ndash689 2008
[14] F-J Santonja R-J Villanueva L Jodar and G Gonzalez-Parra ldquoMathematical modelling of social obesity epidemic inthe region of Valencia Spainrdquo Mathematical and ComputerModelling of Dynamical Systems vol 16 no 1 pp 23ndash34 2010
[15] F J Santonja and L Shaikhet ldquoProbabilistic stability analysisof social obesity epidemic by a delayed stochastic modelrdquoNonlinear Analysis Real World Applications vol 17 pp 114ndash1252014
[16] D Tudor ldquoA deterministic model for herpes infections inhuman and animal populationsrdquo SIAM Review vol 32 no 1pp 136ndash139 1990
[17] A M Lyapunov The general problem of the stability of motionTaylor amp Francis London England 1992
[18] C Vargas-De-Leon ldquoLyapunov functions for two-species coop-erative systemsrdquo Applied Mathematics and Computation vol219 no 5 pp 2493ndash2497 2012
[19] D W Jordan and P Smith Nonlinear Ordinary DifferentialEquations An Introduction for Scientists and Engineers OxfordUniversity Press New York NY USA 4th edition 2007
[20] C M Kribs-Zaleta and J X Velasco-Hernandez ldquoA simplevaccination model with multiple endemic statesrdquoMathematicalBiosciences vol 164 no 2 pp 183ndash201 2000
[21] S Barquera I Campos-Nonato L Hernandez-Barrera APedroza-Tobıas and J A Rivera-Dommarco ldquoPrevalence ofobesity in mexican adults ensanut 2012rdquo Salud Publica deMexico vol 55 supplement 2 pp S151ndashS160 2013
Perhaps it is possible to build amore general nonlinearmodelwhere in the case in which the contribution due to nonsocialeffects gradually fades the backward bifurcation of the Ejimaet al model could be obtained as a limit case
Competing Interests
The authors declare that they have no conflict of interests
Acknowledgments
This research was supported by the UACM-SECITI ResearchGrant PI-2013-34 Cruz Vargas-De-Leon would like toexpress his indebtedness to Dr Ramon Reyes Carreto for hishospitality during his stay as visiting professor at the UAGroin Chilpancingo
References
[1] World Health Organization Obesity Preventing and Managingthe Global Epidemic Report of a WHO Consultation WHOTechnical Report Series No 894 WHO Geneva Switzerland2000 httpswwwgooglecommxurlsa=tamprct=jampq=ampesrc=sampsource=webampcd=1ampved=0ahUKEwjlno 3tsDRAhVqi1QKH-W0rBPIQFggZMAAampurl=http3A2F2Fwhqlibdocwhoint2Ftrs2FWHO TRS 894pdfampusg=AFQjCNFLusNXrJz-xFKX1xpQwkHLEr-Lohgampbvm=bv144224172damcampcad=rja
[2] C L Vanvrancken-Tompkins andMaM Sothern ldquoPrevencionde la obesidad en los ninos desde el nacimiento hasta loscincoanosrdquo in Enciclopedia Sobre el Desarrollo de la PrimeraInfancia R Tremblay R Barr R Peters and M Boivin Edsvol 1ndash7 Centre of Excellence for EarlyChildhoodDevelopmentQuebec Canada 2010
[3] S K Garg H Maurer K Reed and R Selagamsetty ldquoDiabetesand cancer two diseases with obesity as a common risk factorrdquoDiabetes Obesity andMetabolism vol 16 no 2 pp 97ndash110 2014
[4] National Heart ldquoLung and Blood Institute World Health WhatAre the Health Risks of Overweight and Obesityrdquo httpswwwnhlbinihgovhealthhealth-topicstopicsoberisks
[5] World Health Organization (WHO) Global Health Observa-tory Data Repository WHO Geneva Switzerland 2011 httpappswhointghodata
[6] World Health Organization The Global Strategy on Diet Phys-ical Activity and Health (DPAS) WHO Geneva Switzerland2004 httpwwwwhointnmhwha59dpasen
[7] N A Christakis and J H Fowler ldquoThe spread of obesity in alarge social network over 32 yearsrdquoThe New England Journal ofMedicine vol 357 no 4 pp 370ndash379 2007
[8] K Ejima K Aihara and H Nishiura ldquoModeling the obesityepidemic social contagion and its implications for controlrdquoTheoretical Biology and Medical Modelling vol 10 article 17 pp1ndash13 2013
[9] A J Arenas G Gonzalez-Parra and L Jodar ldquoPeriodic solu-tions of nonautonomous differential systems modeling obesitypopulationrdquo Chaos Solitons amp Fractals vol 42 no 2 pp 1234ndash1244 2009
[10] G Gonzalez-Parra L Acedo R-J Villanueva Mico and A JArenas ldquoModeling the social obesity epidemic with stochasticnetworksrdquo Physica A Statistical Mechanics and its Applicationsvol 389 no 17 pp 3692ndash3701 2010
[11] G Gonzalez-Parra L Jodar F J Santonja and R J VillanuevaldquoAn age-structured model for childhood obesity mathematicalpopulation studiesrdquo An International Journal of MathematicalDemography vol 17 no 1 pp 1ndash11 2010
[12] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 Article ID e1000968 2010
[13] L Jodar F J Santonja and G Gonzalez-Parra ldquoModelingdynamics of infant obesity in the region of Valencia SpainrdquoComputers andMathematics with Applications vol 56 no 3 pp679ndash689 2008
[14] F-J Santonja R-J Villanueva L Jodar and G Gonzalez-Parra ldquoMathematical modelling of social obesity epidemic inthe region of Valencia Spainrdquo Mathematical and ComputerModelling of Dynamical Systems vol 16 no 1 pp 23ndash34 2010
[15] F J Santonja and L Shaikhet ldquoProbabilistic stability analysisof social obesity epidemic by a delayed stochastic modelrdquoNonlinear Analysis Real World Applications vol 17 pp 114ndash1252014
[16] D Tudor ldquoA deterministic model for herpes infections inhuman and animal populationsrdquo SIAM Review vol 32 no 1pp 136ndash139 1990
[17] A M Lyapunov The general problem of the stability of motionTaylor amp Francis London England 1992
[18] C Vargas-De-Leon ldquoLyapunov functions for two-species coop-erative systemsrdquo Applied Mathematics and Computation vol219 no 5 pp 2493ndash2497 2012
[19] D W Jordan and P Smith Nonlinear Ordinary DifferentialEquations An Introduction for Scientists and Engineers OxfordUniversity Press New York NY USA 4th edition 2007
[20] C M Kribs-Zaleta and J X Velasco-Hernandez ldquoA simplevaccination model with multiple endemic statesrdquoMathematicalBiosciences vol 164 no 2 pp 183ndash201 2000
[21] S Barquera I Campos-Nonato L Hernandez-Barrera APedroza-Tobıas and J A Rivera-Dommarco ldquoPrevalence ofobesity in mexican adults ensanut 2012rdquo Salud Publica deMexico vol 55 supplement 2 pp S151ndashS160 2013
