key: cord-0134182-5q7lxkpi
authors: Crokidakis, Nuno; Sigaud, Lucas
title: Role of inflexible minorities in the evolution of alcohol consumption
date: 2022-04-08
journal: nan
DOI: nan
sha: c9b7c75c616e81e1cfa2865d936ee72e29ca3506
doc_id: 134182
cord_uid: 5q7lxkpi

In this work we study a simple mathematical model for drinking behavior evolution. For this purpose, we considered three compartments, namely Susceptible individuals $S$ (nonconsumers), Moderated drinkers $M$ and Risk drinkers $R$. Inside the $S$ and $R$ compartments, we considered the presence of inflexible or zealot agents, i.e., individuals that never change their behavior (never drink or always drink a lot). These inflexible agents are described by fixed densities $s_I$ and $r_I$, for nonconsumer inflexible and risk drinking inflexible individuals, respectively. We analyze the impact of the presence of such special agents in the evolution of drinking behavior in the population. Specifically, since the presence of inflexible agents are similar to the introduction of quenched disorder in the model, we are interested in the impact of such disorder in the critical behavior of the system. Our analytical and numerical results indicate that the presence of only one class of inflexible agents, $s_I$ or $r_I$, destroys one of the two possible absorbing phases that are observed in the model without such inflexibles, i.e., for $s_i=r_I=0$. In the presence of the both kinds of inflexible agents simultaneously, there are no absorbing states anymore. Since absorbing states are collective macroscopic states with the presence of only one kind of individuals in the population, nonconsumers or risk drinkers, we argue that the inclusion of inflexible agents in the population makes the model more realistic. In addition, the work makes a contribution to studies on the impact of quenched disorder in nonequilibrium phase transitions, that are a subject of interest for Nonequilibrium Statistical Physics.

A myriad of contagion processes have been studied by epidemic models that go far beyond the scope of infectious diseases [1] and into the very diverse area of social behavior dynamics, such as corruption [2] , cooperation [3] , obesity [4] , ideological conflicts [5] , fanaticism [6] , rumor spreading [7] , etc. The applications for such models also include general populational behavior, such as rising (and falling) of ancient empires [8] , violence [9] , radicalization [10] and tax evasion dynamics [11, 12] , andof particular interest to this work -addiction.

Regarding addiction, epidemic models have been used previously to describe the populational consumption evolution of different types of substances, from tobacco [13] and alcohol [14] to heavier drugs such as cocaine [15] and heroine [16] . Although addiction is very often seen and treated as an individual's condition, social interaction with people that are more or less users of the respective substance may influence the individual's reaction and consumption, by peer pressure, condemnation or positive reinforcement, ultimately changing their consumption levels. Therefore, it can be treated like a contagion process mediated by the social interactions of individuals with different degrees of addiction. Since alcohol, in particular, is one of the most socially accepted addictive substances for consumption and commerce, as well as one in which social peer pressure can be most effective [17] , it is a prime candidate to be modelled by epidemic models [18, 19] , specially due to the increase of alcohol consumption not only by social interactions but also spontaneously, which has been documented as a consequence of depression, isolation and even the COVID-19 pandemics [20, 21, 22, 23] .

Recently, we addressed the evolution of alcoholism as an epidemic using a compartment contagion model [24] that subdivided the adult population into three groups, following the alcohol consumption categorization of the World Health Organization [25] , namely nonconsumers, social (or moderate) consumers and excessive (or risk) consumers. We treated the ensemble as a fully-connected population subdivided into three groups that could influence individuals from one another as well as spontaneously migrate from social to excessive drinkers.

In this work, we study the influence of inflexible individuals in both nonconsumer and excessive consumer groups. Inflexible, or zealot, individuals are the ones that never change their status -in this case, either nonconsumers that never acquire the habit of drinking or excessive drinkers that never diminish their alcohol consumption. This is analogous to introducing disorder in the system, considerably altering its critical behavior. This has been studied before in many opinion-related contagion modelling [2, 26, 27, 28, 29] , and we are interested in understanding its impact on the alcohol-consumption model structure of our previous work on the subject [24] .

Thus, we describe the modelling itself in the next section, along with a brief outline of previous models on alcohol consumption. In Section 3, we present and discuss the results obtained with the new model with inflexibles, considering the presence of either i) inflexible nonconsumers; ii) inflexible excessive consumers; or iii) both types of inflexibles. We follow that with some final remarks and conclusions obtained in this work, and we also include at the end appendixes with the bulk of the mathematical analytical deductions.

Our model is based on the proposal of Refs. [14, 18, 19, 24, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45] that treat alcohol consumption as a disease that spreads by social interactions. In short, we consider a population of N individuals, which is divided in 3 compartments [24] , namely:

• S: nonconsumer individuals, individuals that have never consumed alcohol or have consumed in the past and quit. In this case, we will call them Susceptible individuals, i.e., susceptible to become drinkers, either again or for the first time;

• M: nonrisk (or moderate) consumers, individuals with regular low consumption. We will call them Moderated drinkers;

• R: risk (or excessive) consumers, individuals with regular high consumption. We will call them Risk drinkers; This simple compartmental model was proposed recently [24] , following many other three-compartmental models for alcohol consumption, albeit with structural differences [14, 19, 30] . Although more complex models incorporated one or more other compartments [35, 36, 37, 38, 39] or even external influence on the mobility between compartments (e.g. the effect of mass media campaigns) [40, 41, 42, 43] , the authors focused only on compartments corresponding to the three main alcohol consumption categories, listed above, according to the World Health Organization [25] , in order to compare with available data [24] . They incorporated elements from models from both Santonja et al. [14] (spontaneous transition) and Gorman et al. [19] (transition mediated by social interaction) and verified the occurrence of two distinct active-absorbing phase transitions [46] : one of the phases presents only S individuals, and the other one presents only R individuals.

Thus, in addition to the interaction rules defined in [24] , we considered here that some agents in the population act as inflexibles or zealots [2, 26, 27, 28, 29, 47, 48, 49, 50] . In such a case, we consider that we have a certain number S I of inflexibles related to nonconsumer individuals, i.e., individuals that have never consumed alcohol and they will keep this behavior during all their lives. In addition, we also considered a certain number R I of inflexibles related to risk drinkers, i.e., individuals that will always consume alcohol in great quantities. In the following subsections we will elaborate upon the motivations to include such kind of inflexible individuals in the population. Thus, the total number of nonconsumer individuals are given by S total = S + S I and the total number of risk drink individuals are given by R total = R + R I , where S and R denote the noninflexible individuals of each class. Since for the moderated drinkers we are not considering inflexible individuals, for this class we have simply M total = M. • R γ → S: a Risk drinker (R) becomes a Susceptible agent (S) with probability γ if he/she is in contact with Susceptible individuals (S and S I ));

A schematic representation of such transitions is shown in Figure 1 .

In this work we consider population homogeneous mixing, i.e., a fully-connected population of N individuals. Notice from the above rules that inflexible individuals cannot change compartment, since their opinions/states are frozen in time. However, they can persuade other noninflexible individuals to change compartments, for example a inflexible Susceptible agent S I can persuade a noninflexible Risk drinker R to stop drinking and change to state S with probability γ.

As discussed in [24] , the rules above are governed by transition probabilities. β represents an "infection" probability, i.e., the probability that a consumer (M or R) individual turns a nonconsumer one (S) into a drinker. The Risk drinkers R or R I can also "infect" the Moderated M agents and turn them into noninflexible Risk drinkers R, which occurs with probability δ. This transition M → R can also occur spontaneously, with probability α, if a given agent increase his/her alcohol consumption. As stated in the introduction, above, the increase of alcohol consumption has been documented to occur under stressful circumstances (like the COVID-19 pandemic [21] ) or clinical depression [20] , regardless of social interaction with Risk drinkers. Finally, the probability γ represents the infection probability that turn noninflexible Risk drinkers R into noninflexible Susceptible agents S. In this case, it can represent the pressure of social contacts (family, friends, etc) over individuals that drink excessively. For simplicity, we did not take into account transitions from Risk (R) to Moderate (M), assuming that, as a rule, once an individual reaches a behavior of excessive consumption of alcohol, contact with Moderate 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 able to exert a higher pressure on them to quit drinking.

In the next subsections we consider three distinct cases, according to inflexibility: (i) there are only S I inflexible individuals in the population (R I = 0); (ii) there are only R I inflexible individuals in the population (S I = 0); (iii) both inflexible individuals S I and R I are present in the population.

In this section we will consider only R I inflexible individuals in the population, i.e., there are some individuals that will always consume alcohol in large quantities. It is documented that many risk drinkers are not open to change their habits or even admit their level of alcohol consumption due to many reasons, such as social stigma [51] , mistrust in governments and/or treatment centers [52] , personal (or family) denial [53] , etc. It is estimated that, in the United States, only about 1.3% of almost 19 million people above 12 years old with a substance addiction disorder admit their condition and seek treatment [54] , and these numbers may have gotten even worse during the COVID-19 pandemics [23] . This issue is so relevant that some models subdivide the risk consummers compartment into admitters and deniers [38] , and we incorporate it here in the form of risk drinkers that cannot be influenced by nonconsummers. Thus, we are not considering in this section the presence of nonconsumer inflexibles, i.e., we have S I = 0.

Let us work with population densities. For a fixed population composed by N individuals, we can define the densities for each time step t, i.e., s(t) = S(t)/N, m(t) = M(t)/N and r(t) = R(t)/N. In addition, we have a fixed density of inflexible risk drinkers r I = R I /N, that does not depend on t. For this case, the normalization condition can be written as

valid at each time step t. For such a case, based on the microscopic rules defined in section 2, one can write the master equations that describe the time evolution of the densities as follows: 

One can start analyzing the time evolution of the compartments s, m and r. We numerically integrated Eqs. (2), (3) and (4) in order to analyze the effects of the variation of the model's parameters. As initial conditions, we considered m(0) = 0.01, r(0) = 0.0 and s(0) = 1 − m(0) − r I . We will consider as illustration the case r I = 0.10, i.e., 10% of the population is composed by inflexible risk drinkers. For simplicity, we fixed α = 0.03, δ = 0.07 and γ = 0.15, for comparison with the case r I = 0 [24] , and we considered some values of the infection probability β. From Fig.  2 we can see that the system evolves to stationary states for sufficient long times. For the case with β = 0.02 ( Fig. 2 (a) ), the population does not fall in an absorbing state as in the case r I = 0 [24] , even for a very small value of β. For β = 0.10 ( Fig. 2 (b) ) we observe the coexistence of the subpopulations s, m, r and r I . Finally, for the case β = 0.20 we can see that, for sufficient long times, the system achieves an absorbing state with s = m = 0 and r + r I = 1. These results will be discussed in more detail below. Now we can analyze the stationary properties of the model in the presence of inflexible risk drinkers r I . We denote the stationary densities as s = s(t → ∞), m = m(t → ∞) and r = r(t → ∞). Some details of the analytical calculations are exhibited in Appendix A. There we can find two possible solutions for the density of noninflexible nonconsumers s: a trivial solution s = 0, and another solution s = 0. If the solution s = 0 is valid, we can also find analytically that m = 0 (see Appendix A). From the normalization condition of Eq. (1), we obtain r = 1 − r I . This solution represents an absorbing state similar to the one observed in the case s I = r I = 0 [24] , i.e., we have the absence of S and M populations, leaving only risk drinkers in the population, with inflexible r I , and noninflexible r, risk drinkers. The first stationary solution for the case r I = 0 is then (s, m, r) = (0, 0, 1 − r I ).

Let us consider the case s = 0. We can see in Appendix A that m = 0 is not a solution if r I = 0. Thus, the second absorbing solution observed for r I = 0, namely (s, m, r) = (1, 0, 0) [24] , is destroyed in the presence of inflexible risk drinkers r I . Despite the theoretical interest of physicists in absorbing phase transitions [55, 56] , from the practical point of view an absorbing phase with no risk drinkers is not realistic. Thus, the presence of R I inflexible individuals makes the model more realistic, despite the presence of the other absorbing state (s, m, r) = (0, 0, 1 − r I ).

An illustration of the stationary states of the model is exhibited in Fig. 3 , where we show the stationary densities as functions of β for fixed α = 0.03, γ = 0.15 and δ = 0.07. For this figure, we consider r I = 0.10, i.e., 10% of the population is formed by inflexible risk drinkers. The results were obtained by the numerical integration of Eqs.

(2) -(4). We can see that for small values of β there is no absorbing phase as in the case r I = 0 (s = 1, m = r = 0 for finite β) [24] . Even for β = 0.0 there is a coexistence among nonconsumers s and inflexible risk drinkers r I . However, for larger values of β (β >≈ 0.13 in Fig. 3) we observe an absorbing phase s = m = 0 and r + r I = 1, as obtained analytically (see Appendix A). For intermediate values of β we observe the coexistence of s, m, r and r I subpopulations. For the considered parameters, this region is 0 < β <≈ 0.13 in Fig. 3 .

In equilibrium systems like magnetic models, the presence of disorder usually destroys phase transitions [57, 58] . Here, we observe a similar effect in a nonequilibrium system. In this case, the frozen states of the inflexible agents work in the model as the introduction of quenched disorder. As in magnetic systems, one can expect that disorder can induce or suppress a phase transition, as was also observed in the kinetic exchange opinion model in the presence of inflexibles [29] . However, the presence of disorder in models with absorbing states does not lead to the destruction of active-absorbing phase transitions, at least in low-dimensional systems [59, 60] . At mean-field level, to the best of our knowledge, the suppression of nonequilibrium phase transitions to absorbing states had not been previously observed. Thus, the model makes a contribution to the study of the impact of quenched disorder in nonequilibrium phase transitions. 

In this section we will consider a more commonly encountered scenario, where there are only S I inflexible individuals in the population, i.e., there are some individuals that have never consumed alcohol and they will keep this behavior during all their lives. Such individuals can be representative, for example, of one of the many religions and/or places that forbid alcohol consumption. However, even outside communities where alcohol consumption is forbidden, persons that absolutely refrain from drinking are becoming more common in many regions of the world [25, 61] , and recent studies show the influence that teetotaller young adults can have in their social environments [62] . Thus, in order to analyze the sole influence of inflexible nonconsummers in the model, we are not considering in this section the presence of inflexible risk drinkers, i.e., we have R I = 0.

As in the previous subsection, we will work with population densities. In such a case, we have a fixed density of inflexible nonconsumers s I = S I /N, that does not depend on t. For this case, the normalization condition can be written as

Thus, based on the microscopic rules defined in section 2, one can write the master equations that describe the time evolution of the densities, Again, one can start by analyzing the time evolution of the compartments s, m and r. We numerically integrated Eqs. (6), (7) and (8) in order to analyze the effects of the variation of the model's parameters. As initial conditions, we considered m(0) = 0.01, r(0) = 0.0 and s(0) = 1 − m(0) − s I . We will consider as a sample case s I = 0.10, i.e., 10% of the population is composed by inflexible nonconsumers. For simplicity, we fixed α = 0.03, δ = 0.07 and γ = 0.15, for comparison with the case s I = 0 [24] , and we considered some values of the infection probability β. From Fig. 4 we can see that the system evolves to stationary states for sufficiently long times. For the case with β = 0.02 ( Fig. 4 (a) ), the population falls in an absorbing state with m = r = 0 and s + s I = 1, as in the case s I = 0 [24] . For β = 0.10 ( Fig. 4 (b) ) we observe the coexistence of subpopulations s, m, r and s I . Finally, for the case β = 1.00 (Fig. 4 (c) ) we can see that, for sufficiently long times, the system does not achieve another absorbing state as in the case s I = 0 [24] , even for β = 1.0. These results will be discussed in more detail in the following.

Let us analyze the stationary states of the model. Some details of the calculations 

In this section we will consider both S I and R I inflexible individuals in the population. This describes the coexistence of both inflexible nonconsumers and inflexible risk drinkers, and we will analyze here the influence on the overall population that these two radical groups exert. This is the more general case, with two distinct sources of disorder in the model.

As before, we will work with population densities. In this case, we have fixed densities of inflexible risk drinkers r I = R I /N and inflexible nonconsumers s I = S I /N, both do not depending on t. For this case, the normalization condition can be written as

Based on the microscopic rules defined in section 2, one can write the master equations that describe the time evolution of the densities,

One can start analyzing the time evolution of compartments s, m and r. We numerically integrated Eqs. (10), (11) and (12) . As initial conditions, we considered m(0) = 0.01, r(0) = 0.0 and s(0) = 1 − m(0) − r I − s I . We will consider as an example the case s I = r I = 0.10, i.e., 10% of the population is formed by inflexible nonconsumers and 10% of inflexible risk drinkers. As in the previous subsections, we fixed α = 0.03, δ = 0.07 and γ = 0.15, and we considered some values of the infection probability β. From Fig. 6 , one can readily see that for all shown values of β (β = 0.02 (a), β = 0.10 (b), and β = 1.00 (c)) there are no absorbing states whatsoever, i.e. there are no trivial solutions for the equilibrium states of the model. On the contrary, the presence of inflexibles are responsible for maintaining all compartments with occupation values different from zero, promoting the social interactions that keep noninflexible individuals changing their drinking status. As should be expected, for low values of the transmission coefficient β (Fig. 6 (a) ) we have a predominance of nonconsumers in the equilibrium state; on the other hand, for high values of β the situation is reversed (Fig. 6 (c) ), with risk drinkers attaining the prominent role, once a transient phase of moderate consumers majority has passed -actually, the noninflexible nonconsumer population rapidly decreases to near zero and the moderate consumer population tends to almost the same value as the s + s I compartment. Intermediate values of β ( Fig. 6 (b)) lead to a more homogeneous mixing of the three consumers group, which is prone to be a more accurate description of most communities where no alcohol-consumption ban exists [25] . An illustration of the stationary states of the model is exhibited in Fig. 7 , where we show the stationary densities as functions of β for fixed α = 0.03, γ = 0.15 and δ = 0.07. For this figure, we consider s I = r I = 0.10, i.e., 20% of the population is formed by inflexible agents, 10% nonconsumers and 10% risk drinkers.

The results of Fig. 7 corroborate the ones described above for Fig. 6 . One can see that only for small values of β a majority of nonconsumers are present for the studied conditions, but as β increases a predominance of risk drinkers quickly supersedes the other subpopulations. Curiously, no stationary solution with s = m = r can be found in the model, which was the same case for the model without the presence of inflexibles (s I = r I = 0) [24] . It should also be noted that the increase of the transmission coefficient β value leads to a fast predominance of risk drinkers due to a combination of factors: as R increases, more individuals migrate from S to M, since β depends directly on the R subpopulation; also, the increase of the number of M individuals leads to a spontaneous migration to the R compartment via the α coefficient -since this mechanism is not available for any other transition pathway, the predominance of risk drinkers is present for a much wider range of β values than nonconsumers, even though they present, in this study, the same amount of inflexible individuals.

The presence of the two kinds of inflexible agents, S I and R I , leads to a system where there are no absorbing transitions anymore. In such a case, as discussed before, the model becomes more realistic, since collective macroscopic states with the absence of at least one of the subpopulations (nonconsumers, moderate drinkers or risk drinkers) are not realistic from the practical point of view.

In this work we study a model that constitutes an expansion of a previously studied simplified contagion model for alcoholism consumption [24] , where the population was subdivided into three compartments, namely nonconsumers, moderate consumers and risk drinkers [25] . In this work, we include a fraction of inflexible nonconsumers and/or risk drinkers, and study the subsequent effects on the stationary behavior of their subpopulations.

The first major difference observed between the previously studied simplified model [24] and the newer more complex one presented here is the destruction of one (when one group of inflexible individuals was included) or both (when both inflexible groups are present) absorbing states. This is analog to other equilibrium systems such as magnetic models, where the presence of disorder usually destroys phase transitions [57, 58] . The introduction of inflexible agents works as the inclusion of quenched disorder in the model, which can suppress a phase transition. However, in this model we observe the suppression of a nonequilibrium phase transition to absorbing states, which had not been previously observed in studies of quenched disorder in nonequilibrium phase transitions. Furthermore, the absence of previously observed absorbing states shows the continual influence of inflexible -or zealot -groups. The presence of only, for example, radical nonconsumers intermixed with the population rapidly leads the whole group to become nonconsummers over time; likewise, the converse could be observed with the presence of only radical risk drinkers. On the other hand, the presence of both radical groups mingled with the noninflexible individuals leads to final stationary states where all groups still have representatives -i.e., no absorbing states are observed. The relative predominance (or not) of an individual group is directly linked to the transmission coefficient values, representing the power of persuasion between the different compartment groups. This also represents a much more realistic approach to describing worldwide alcohol consumption, since -with perhaps the exception of some communities where alcohol consumption is a criminal offense -nonconsumers, moderate drinkers and risk drinkers alike are present in the society.

As future extensions, the model can be considered in regular lattices, as well as complex networks. In lattices or networks, in addition to the present dynamics of the model studied here, we can consider diluted structures, i.e., lattices/networks with empty sites. In such a case, we can consider mobility of the agents through the network, that can lead to rich phenomenology [63] . In addition, other kinds of special agents like contrarians [64, 65, 66, 67] should also be considered.

This last equation presents two solutions, namely s = 0 and r = β(m + r I ) γ − β (A.1)

If s = 0 solution, the limit t → ∞ in Eq. (3) leads to a solution m = 0. From the normalization condition, Eq. (1), we found r = 1 − r I . This set (s, m, r) = (0, 0, 1 − r I ) represents the absorbing phase found in section 3.1 (See also Fig. 3) .

On the other hand, if the valid solution is s = 0, Eq. (A.1) is valid, but we have also to consider the long-time limit in another equation. Taking the limit t → ∞ in Eq. This second order polynomial also indicates that m = 0 is not a solution for the system with r I = 0. Such polynomial gives us the stationary solution for m. Substituting the result in Eq. (A.3), we can find the stationary solution for s. Finally, the stationary solution for r can be found from the normalization condition, Eq. (1). This set (s, m, r) represents the solution for the coexistence phase (see the discussion in section 3.1 and Fig. 3) . where the function C(r) is given by C(r) = β α + (δ + γ) r [δ 2 r 3 + 2(α + γ) δ r 2 + (α + γ) 2 

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The authors thank Serge Galam for some suggestions. Financial support from the Brazilian scientific funding agencies CNPq (Grants 310893/2020-8 and 311019/2017-0) and FAPERJ (Grant 203.217/2017) is also acknowledged.

Appendix A. Analytical considerations: Model in the presence of R I inflexible individualsTaking the limit t → ∞ in Eq. (2), we obtain (−β m − β(r + r I ) + γ r) s = 0, where we denoted the stationary fractions as s = s(t → ∞), m = m(t → ∞) and r = r(t → ∞).