Modeling the evolution of drinking behavior: A Statistical Physics perspective
aa r X i v : . [ phy s i c s . s o c - ph ] A ug Modeling the evolution of drinking behavior: AStatistical Physics perspective
Nuno Crokidakis, Lucas Sigaud
Instituto de F´ısica, Universidade Federal Fluminense, Niter´oi, Rio de Janeiro, Brazil
Abstract
In this work we study a simple compartmental model for drinking behaviorevolution. The population is divided in 3 compartments regarding their al-cohol consumption, namely Susceptible individuals S (nonconsumers), Mod-erated drinkers M and Risk drinkers R. The transitions among those statesare rules by probabilities. Despite the simplicity of the model, we observedthe occurrence of two distinct nonequilibrium phase transitions to absorbingstates. One of these states is composed only by Susceptible individuals S,with no drinkers ( M = R = 0). On the other hand, the other absorbingstate is composed only by Risk drinkers R ( S = M = 0). Between thesetwo steady states, we have the coexistence of the three subpopulations S, Mand R. Comparison with abusive alcohol consumption data for Brazil showsa good agreement between the model’s results and the database. Keywords:
Dynamics of social systems, Alcoholism model, Collectivephenomena, Nonequilibrium Phase Transitions, Absorbing states
1. Introduction
Epidemic models have been widely used to study contagion processes suchas the spread of infectious diseases [1] and rumors [2]. This kind of modelhas also been used for the spread of social habits, such as the smoking habit[3], cocaine [4] and alcohol consumption [5], obesity [6], corruption [7], coop-eration [8], ideological conflicts [9], and also to other problems like rise/fallof ancient empires [10], dynamics of tax evasion [11] and radicalization phe-nomena [12].
Email address: [email protected] (Nuno Crokidakis)
Preprint submitted to Elsevier August 26, 2020 he main reason such social behaviors can be modelled by contagion pro-cesses is the response by elements of the ensemble to the social context ofthe studied subject. Both social or peer pressure and positive reinforcementfrom other agents, regardless if the behavior brings positive or negative con-sequences to the individual, can influence each one’s way of life. Therefore,models for the epidemics of infectious diseases are also able to describe thespread of such tendencies, like alcoholism [13, 14].The standard medical way of categorizing alcohol consumption [15] isin three groups - nonconsumers, moderate (or social) consumers and risk(or excessive) consumers; thus, modeling of the interactions and consequentchanges of an individual from one group to another is governed by interactionparameters. One interesting aspect that should be taken into considerationwhen modeling alcohol consumption is the tendency on some individuals togradually increase their consumption rate, not due to social susceptibility,but when under stressful or depressing circumstances, since alcohol plays amajor role both as cause and consequence of depression, for instance [16].This means that one can attach a probability of a moderate drinker to be-come an excessive drinker that is dependent only on the actual moderatedrinkers population size, instead of the two population groups involved inthe change. If one considers the current world situation with the recentcoronavirus disease 2019 outbreak (COVID-19), this self-induced increase inalcohol consumption is not only realistic, but also becomes more prominent- this has been observed in a myriad of studies this year detailing the con-sequences and dangers of both alcohol withdrawal (in places where it hasbecome harder to legally acquire alcohol during the pandemic) and alcoholconsumption increase [17, 18, 19, 20].This work is organized as follows. In Section 2, we present the modeland define the microscopic rules that lead to its dynamics. The analyticaland numerical results are presented in Section 3, including comparisons withBrazil’s alcohol consumption data for a range of eleven years, used as a casestudy in order to evaluate the present model. Finally, our conclusions arepresented in section 4. 2 . Model
Our model is based on the proposal of references [5, 13, 14, 21, 22, 23,24, 25] that treat alcohol consumption as a disease that spreads by socialinteractions. In such case, we consider an epidemic-like model where thetransitions among the compartments are governed by probabilities. In thiswork we consider population homogeneous mixing, i.e., a fully-connectedpopulation of N individuals. This population is divided in 3 compartments,namely: • S : nonconsumer individuals, individuals that have never consumed al-cohol or have consumed in the past and quit. In this case, we willcall them Susceptible individuals, i.e., susceptible to become drinkers,either again or for the first time; • M : nonrisk consumers, individuals with regular low consumption. Wewill call them Moderated drinkers; • R : risk consumers, individuals with regular high consumption. We willcall them Risk drinkers;To be precise, a moderated drinker is a man who consumes less than 50cc of alcohol every day or a woman who consumes less than 30 cc of alcoholevery day. On the other hand, a risk drinker is a man who consumes morethan 50 cc of alcohol every day or a woman who consumes more than 30cc of alcohol every day [5] . Since we are considering a contagion model,the probabilities related to changes in agents’ compartments represent thepossible contagions. The transitions among compartments are as following: • S β → M : a Susceptible agent ( S ) becomes a Moderated drinker ( M )with probability β if he/she is in contact with Moderated ( M ) or Risk( R ) drinkers; • M α → R : a Moderated drinker ( M ) becomes a Risk drinker ( R ) withprobability α ; Alternatively, Brazil’s Ministry of Health also includes people with frequent episodesof excessive drinking - namely 100 cc for men and 80 cc for women in one occasion - to beRisk drinkers [26]. M δ → R : a Moderated drinker ( M ) becomes a Risk drinker ( R ) withprobability δ if he/she is in contact with Risks drinkers ( R ); • R γ → S : a Risk drinker ( R ) becomes a Susceptible agent ( S ) withprobability γ if he/she is in contact with Susceptible individuals ( S );In the above rules, β represents an “infection” probability, i.e., the prob-ability that a consumer (M or R) individual turns a nonconsumer one intodrinker. The Risk drinkers R can also “infect” the Moderated M agents andturn them into Risk drinkers R, which occurs with probability δ . These twoinfections occur by contagion, in our model, where individuals belonging to agroup with a higher degree of consumption can influence others to drink morevia social contact. This transition M → R can also occur spontaneously, withprobability α , if a given agent increase his/her alcohol consumption - this isthe only migration pathway from one group to another, in this model, thatdoes not depend on the population of the receiving compartment, since itcorresponds to a self-induced progression from Moderate ( M ) to Risk ( R )drinking. As stated in the introduction, above, the increase of alcohol con-sumption has been documented to occur under stressful circumstances (likethe COVID-19 pandemic) or clinical depression, regardless of social interac-tion with Risk drinkers. Finally, the probability γ represents the infectionprobability that turn Risk drinkers R into Susceptible agents S . In this case,it can represent the pressure of social contacts (family, friends, etc) over in-dividuals that drink excessively. We did not take into account transitionsfrom Risk ( R ) to Moderate ( M ), assuming that, as a rule, once an indi-vidual reaches a behavior of excessive consumption of alcohol, contact withModerate drinkers does not imply on a tendency to lower one’s consumption- meanwhile, it is assumed that contacts that do not drink at all are ableto exert a higher pressure on them to quit drinking. It is not that the fromRisk to Moderate transition cannot occur - it is just that for our model thisprobability, when comparing it with the overall picture, is negligible.For simplicity, we consider a fixed population, i.e., at each time step t wehave the normalization condition s ( t ) + m ( t ) + r ( t ) = 1, where we definedthe population densities s ( t ) = S ( t ) /N , m ( t ) = M ( t ) /N and r ( t ) = R ( t ) /N .Since we will only deal with the relative proportions among the three differentgroups in relation to the total population N , i.e. the population densities,we will not take into account birth-mortality relations and populational in-crease/decrease effects. So, even if N is not a constant number, for all mod-4lling purposes it will not matter due to the fact that we will deal only withthe s ( t ), m ( t ) and r ( t ) subpopulations in relation to the total population.One other way of looking at this approximation is to consider only the adultpopulation as relevant to our modelling, and assume that new individualscoming of age correspond to the number of deaths [24, 25].
3. Results
Based on the microscopic rules defined in the previous subsection, one canwrite the master equations that describe the time evolution of the densities s ( t ), m ( t ) and r ( t ) as follows, ds ( t ) dt = − β s ( t ) m ( t ) − β s ( t ) r ( t ) + γ s ( t ) r ( t ) , (1) dm ( t ) dt = β s ( t ) m ( t ) + β s ( t ) r ( t ) − δ m ( t ) r ( t ) − α m ( t ) , (2) dr ( t ) dt = α m ( t ) + δ m ( t ) r ( t ) − γ s ( t ) r ( t ) , (3)and we also have the normalization condition s ( t ) + m ( t ) + r ( t ) = 1 , (4)valid at each time step t .First of all, one can analyze the early evolution of the population, forsmall times. Considering the initial conditions s (0) ≈ m (0) ≈ /N and r (0) = 0, one can linearize Eq. (2) to obtain dm ( t ) dt = ( β − α ) m ( t ) , (5)that can be directly integrated to obtain m ( t ) = m e α ( R − t , where m = m ( t = 0) and one can obtain the expression for the basic reproduction num-ber R = βα . (6)As it is usual in epidemic models [1, 27], the disease (alcoholism) will persistin the population if R >
1, i.e., for β > α .One can start analyzing the time evolution of the three classes of indi-viduals. We numerically integrated Eqs. (1), (2) and (3) in order to analyze5
50 100 150 200 250 300 350
Time D e n s iti e s smr (a) Time D e n s iti e s smr (b) Time D e n s iti e s smr (c) Time D e n s iti e s smr (d) Figure 1: (Color online) Time evolution of the three densities of agents s ( t ), m ( t ) and r ( t ), based on the numerical integration of Eqs. (1) - (3). The fixed parameters are α = 0 .
03 and δ = 0 .
07, and we varied the parameters β and γ : (a) β = 0 . , γ = 0 . β = 0 . , γ = 0 .
15, (c) β = 0 . , γ = 0 .
30, (d) β = 0 . , γ = 0 .
15. From Eq. (6), weobtain R ≈ .
33 for the panels (a)-(c) and R ≈ .
67 for panel (d). the effects of the variation of the model’s parameters. As initial conditions,we considered s (0) = 0 . m (0) = 0 .
01 and r (0) = 0, and for simplicity wefixed α = 0 .
03 and δ = 0 .
07, varying the parameters β and γ . In Fig. 1 (a),(b) and (c) we exhibit results for fixed β = 0 .
07 and typical values of γ . Onecan see that the increase of γ causes the increase of s and the decrease of m and r . Remembering that γ models the persuasion of nonconsumers S in thesocial interactions with risk drinkers R , i.e., the social pressure of individualsthat do not consume alcohol over their contacts (friends, relatives, etc) thatconsume too much alcohol. On the other hand, in Fig. 1 (d) we considered γ = 0 .
15 and β = 0 .
07. For this case, where we have β > γ , we see that thedensities evolve in time, and in the steady states we observe the survival of6nly the risk drinkers, i.e., for t → ∞ we have s = m = 0 and r = 1. Thislast result will be discussed in more details analytically in the following.As we observed in Fig. 1, the densities s ( t ), m ( t ) and r ( t ) evolve intime, and after some time they stabilize. In such steady states, the timederivatives of Eqs. (1) - (3) are zero. In the t → ∞ limit, Eq. (1) givesus ( − β m − β r + γ r ) s = 0, where we denoted the stationary values as s = s ( t → ∞ ), m = m ( t → ∞ ) and r = r ( t → ∞ ). This last equation hastwo solutions, one of them is s = 0 and from the other solution we can obtaina relation between r and m , r = βγ − β m . (7)Considering now the limit t → ∞ in Eq. (2), one obtains β s r = ( α + δ r − β s ) m . (8)If the obtained solution s = 0 is valid, this relation gives us m = 0 andconsequently from (4) we have r = 1. This solution represents an absorbingstate [28, 29], since the dynamics becomes frozen due to the absence of S and M agents. We will discuss this solution in more details in the following.Considering now the relation (7) and the normalization condition (4), onecan obtain s = 1 − γγ − β m . (9)Substituting (9) and (7) in (8) one obtains 2 solutions, m = 0 and m = [ γ β − α ( γ − β )] ( γ − β ) β [ δ ( γ − β ) + γ ] . (10)Considering this result (10) in Eqs. (9) and (7) we obtain, respectively s = 1 − γ [ γ β − α ( γ − β )] β [ δ ( γ − β ) + γ ] , (11) r = γ β − α ( γ − β ) δ ( γ − β ) + γ . (12)The obtained eqs. (10) - (12) represent a second possible steady state solutionof the model, that is a realistic solution since the three fractions s , m and r coexist in the population. 7e can look to Eq. (10) in more details. It can be rewritten in the criticalphenomena perspective as [30, 31] m = ( α + γ ) ( β − β (1) c ) ( β (2) c − β ) β [ δ ( γ − β ) + γ ] , (13)or in the standard form m ∼ ( β − β (1) c ) ( β (2) c − β ), where β (1) c = α γ/ ( α + γ )and β (2) c = γ . Thus, considering the density m as a kind of order parameter,one observe in this model two distinct nonequilibrium phase transitions. Thesolution (10) is valid in the range β (1) c < β < β (2) c . Notice that one can rewriteEq. (12) as r ∼ ( β − β (1) c ). In this case, one conclude that for β < β (1) c thesolutions (10) - (12) are not valid, since m < r <
0. Thus, in thisregion β < β (1) c the valid solution is m = r = 0 and from the normalizationcondition we have s = 1. This last solution represents a second absorbingstate, distinct from the first one obtained previously (where s = m = 0 and r = 1). Regarding this first absorbing state, it is valid in the region β > β (2) c .Summarizing, the general solutions are: s = if β < β (1) c − γ [ γ β − α ( γ − β )] β [ δ ( γ − β )+ γ ] if β (1) c < β < β (2) c if β > β (2) c (14) m = if β < β (1) c [ γ β − α ( γ − β )] ( γ − β ) β [ δ ( γ − β )+ γ ] if β (1) c < β < β (2) c if β > β (2) c (15) r = if β < β (1) cγ β − α ( γ − β ) δ ( γ − β )+ γ if β (1) c < β < β (2) c if β > β (2) c (16)where the critical points are given by β (1) c = α γα + γ , (17) β (2) c = γ . (18)8 β S t a ti ona r y D e n s iti e s smr Figure 2: (Color online) Stationary densities s , m and r as functions of the probability β for α = 0 . γ = 0 .
15 and δ = 0 .
07. The lines were obtained from Eqs. (14) - (16).For the considered parameters, the critical points are β (1) c = 0 .
025 and β (2) c = 0 .
15. It isimportant to note that these behaviors are present also for other parameter values, andwhat is shown here works as a pattern.
Based on the results above, we plot in Fig. 2 the steady state values ofthe three densities s , m and r as functions of β . For this graphic, we fixedthe parameters α = 0 . , δ = 0 .
07 and γ = 0 .
15. For such values, we have β (1) c = 0 .
025 and β (2) c = 0 .
15. As discussed previously, for β < .
025 thesystem is in one of the absorbing states in the long-time limit, where thereare only nonconsumer agents in the population, i.e., s = 1 and m = r = 0.For β > .
15 the system becomes frozen in the other absorbing phase, wherethere are only risk drinkers in the population after a long time, i.e., r = 1and s = m = 0. Among those states, we have a realistic region where allthe three kinds of individuals, nonconsumers, moderate drinkers and riskdrinkers coexist in the population.The competition among the contagions cause the occurrence of such threeregions in the model. From one side we have drinkers (moderated and risk)influencing nonconsumers to consume alcohol, with probability β . On theother hand, we have the social pressure of nonconsumers over risk drinkers,with probability γ , in order to make such alcoholics to begin treatment and9 γ β III II I
Figure 3: (Color online) Phase diagram of the model in the plane β vs γ for δ = 0 . α = 0 .
03. The full line represents the critical point β (1) c given by Eq. (17) and thedashed line the other critical point β (2) c given by Eq. (18). I denotes the region wherethe system falls in the absorbing phase s = 1 , m = r = 0, II denotes the region where thethree densities s , m and r coexist and III denotes the region where the system falls in theother absorbing phase s = m = 0 , r = 1. stop drinking. Finally, it is important to mention the parameter α , thatdrives the only transition of the model that does not depend on a direct so-cial interaction. That parameter models the spontaneous increase of alcoholconsumption, and it is also responsible for the first phase transition (togetherwith γ ), since we have β (1) c = 0 for α = 0. It means that the alcohol consump-tion (the ”disease”) cannot be eliminated of the population after a long timeif there is a spontaneous increase of alcohol consumption from individualsthat drink moderately, which is a realistic feature of the model.For clarity, we exhibit in Fig. 3 the phase diagram of the model in theplane β versus γ , separating the three above discussed regions. In Fig. 3,the absorbing phase with s = 1 and m = r = 0 is located in region I for β < β (1) c , the coexistence phase is denoted by II for β (1) c < β < β (2) c (wherethe three densities coexist) and the other absorbing phase where s = m = 0and r = 1 is located in region III. From this figure we see the mentionedcompetition among the contagions. Indeed, if β is sufficiently high, many10onconsumers become moderated drinkers. Such moderated drinkers willbecome risk drinkers (via probabilities α and δ ), and in the case of small γ we will observe after a long time the disappearance of nonconsumers andmoderated drinkers (region III). In the opposite case, i.e., for high γ andsmall β , the flux into the compartment S is intense, and in the long-timelimit the other two subpopulations M and R disappears (region I). Finally, forintermediate values of β and γ the competition among the social interactionslead to the coexistence of the three subpopulations in the stationary states(region II). It is worthwhile to mention that the sizes of regions I and II aredirectly dependent on probability α , while region III is always fixed due toEq. (18). This means that, if parameter α is increased, region I will becomegradually larger, which is an indication that the spontaneous evolution frommoderate to risk drinking behavior increases the latter’s absorbing state. Inconsequence, since probability α represents a percentage of moderate drinkersthat become risk drinkers without the need for social interaction, it is a crucialfactor not only to implement the theoretical model but also to identify apossible percentage of the population that has a natural tendency to presentexcessive alcohol consumption behavior, regardless of their social interactionnetwork. For Fig. 3, for instance, this value is 3%. Larger values of α narrowthe set of parameters that can be chosen in order to realistically describe areal system.Finally, we compare the model’s results with data of drinking abusiveconsumption in Brazil [26]. Data were collected from 2009 to 2019, thusin Fig. 4 the initial time t = 0 represents the fraction of abusive drinkersfor 2009, t = 1 represents the fraction for 2010, and so on. Since the datais for the fraction of people that consume alcohol abusively, we plot thedensity of risk drinkers r ( t ) together with the data. In order to comparethem with the model, we considered for the initial density of risk drinkers r (0) = 0 .
185 and numerically integrated Eqs. (1) - (3). The value 0 .
185 waschosen since it is the fraction of abusive drinkers for 2009 obtained from thedatabase [26]. In addition, we rescaled the time of the simulation results tomatch the time of real data: such simulation time was multiplied by 0 . β = 0 . γ = 0 . α = 0 .
047 and δ = 0 .
2, which indicatethat the probability of finding an individual that will spontaneously becomea risk drinker in Brazil during the last decade is around 4,7%. Furthermore,looking at Eqs. (17) and (18), it is easy to see that in order to model Brazil’s11 t D e n s it y o f r i s k d r i n ke rs modeldata Figure 4: (Color online) Comparison between data of abusive alcohol consumption inBrazil through time and the time evolution of the density of risk drinkers r ( t ) given bythe numerical integration of Eqs. (1) - (3). Parameters are β = 0 . γ = 0 . α = 0 . δ = 0 . population we have β (1) c = 0 .
033 and β (2) c = 0 .
11, which is in accordance withthe relation β (1) c < β < β (2) c , showing that the model describes the availabledata in its most realistic spectrum (region II of Figure 3). Naturally, incomparison with actual data, models should always present the three differentphases, i.e. coexistence between the three different population groups, sincedescriptions with only nonconsumers or risk drinkers are unrealistic. Thisqualitative agreement with Brazil’s database in the realistic spectrum of themodel points to a good, albeit simplistic, modelling.
4. Final remarks
In this work, we have studied a compartmental model that aims to de-scribe the evolution of drinking behavior in an adult population. We con-sidered a fully-connected population that is divided in three compartments,namely Susceptible individuals S (nonconsumers), Moderated drinkers Mand Risk drinkers R. The transitions among the compartments are ruledby probabilities, representing the social interactions among individuals, aswell as spontaneous decisions, in particular from moderate evolving into risk12rinkers, and we studied the model through analytical and numerical calcu-lations.From the theoretical point of view, the model is of interest of Statisti-cal Physics since we observed the occurrence of two distinct nonequilibriumphase transitions. These transitions separate the model in three regions: (I)existence of nonconsumers only; (II) coexistence of the three compartmentsand (III) existence of risk drinkers only. Regions I and III represent two dis-tinct absorbing phases, since the system becomes frozen due to the existenceof only one subpopulation for each case - this means that, in order to de-scribe real populational systems, the parameters must be chosen so that themodel falls in region II, since populations consisting solely of nonconsumersor risk drinkers do not represent a realistic entity. The critical points of suchtransitions were obtained analytically. A comparison with available data forBrazil’s extreme alcohol consumption for the past decade shows a good qual-itative agreement with the model, with the chosen parameters framed withinits realistic boundaries. It will be important in a couple of years time tore-evaluate these results in the light of new data comprising years 2020 and2021, in order to verify the direct effects of the COVID-19 pandemic in theBrazilian population’s alcohol consumption. An hypothesis to be tested is apossible increase in parameter α combined with a corresponding decrease inthe other parameters, corresponding to social interactions.The phase transitions observed in the model are active-absorbing phasetransitions, and the predicted critical exponent for the order parameter is 1( m ∼ ( β − β c ) ) as in the mean-field directed percolation, that is the prototypeof a phase transition to an absorbing state [30, 31]. It would be interestingto estimate numerically other critical exponents of the model, as well as tosimulate it in regular d-dimensional lattices (e.g. square and cubic) in orderto obtain all the critical exponents. This is important to define precisely theuniversality class of the model, as well as its upper critical dimension. Thisextension is left for a future work. Furthermore, it can also be consideredthe inclusion of heterogeneities in the population, like agents’ conviction [32],time-dependent transition rates [33], inflexibility [34], etc. Acknowledgments
The authors thank Ronald Dickman for some suggestions. Financial sup-port from the Brazilian scientific funding agencies CNPq (Grants 303025/2017-13 and 311019/2017-0) and FAPERJ (Grant 203.217/2017) is also acknowl-edge.
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