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International Journal of Drug Policy 22 (2011) 247– 251

Contents lists available at ScienceDirect

International Journal of Drug Policy

jo ur n al homep age: www.elsev ier .com/ locate /drugpo

esearch paper

nalysing the Spanish smoke-free legislation of 2006: A new method to quantifyts impact using a dynamic model

rancisco Guerreroa,∗, Francisco-José Santonjab, Rafael-Jacinto Villanuevac

Department Matemática Aplicada, Facultad de Matemáticas, Universidad de Valencia, Avda. Doctor Moliner 50, 46100 Burjasot, Valencia, SpainDepartment Estadística e Investigación Operativa, Universidad de Valencia, Burjasot, Valencia, SpainInstituto de Matemática Multidisciplinar, Universidad Politécnica de Valencia, Valencia, Spain

r t i c l e i n f o

rticle history:eceived 5 January 2011eceived in revised form 24 March 2011ccepted 3 May 2011

eywords:obacco smokingimulation modelealth policy evaluation

a b s t r a c t

Background: There are many models that study aspects of smoking habits: the influence of price, tax,relapse time, and the effects of prohibition. There are also studies examining the effects of the Spanishsmoke-free law. We wanted to build a model able to separate the effect of the law from the pre-lawevolution of smoking habits.Methods: Using data from the Spanish Ministry of Health and Social Policy, we developed a dynamic modelof tobacco use. The model projects the evolution over time of the number of non-smokers, smokers andex-smokers before 2006. Then, we compared the predictions of the model with data for the years afterthe law came into force, 2006 and 2009.Results: We show that smoke-free law has had a significant impact on different sub-populations. Thenumber of ex-smokers increased significantly in 2006 and this increase was maintained in 2009. Thenumber of smokers also decreased significantly in 2006, but in 2009 this returned to its value before thelaw. Simultaneously, the number of non-smokers decreased in 2009.Conclusions: When the law came into force (2006), its restriction on smoking in public and work places

made many smokers decide to give up smoking, decreasing the number of smokers and increasing thenumber of ex-smokers. In 2009, the majority of those who succeeded in giving up smoking did not returnto the habit. However, the smoke-free law had no effect on new smokers and the number of smokersreturned to previous levels, whereas the number of non-smokers decreased. Therefore, we can concludethat the law had a very positive effect in the first few years but this has dissipated over time, with theexception of ex-smokers, whose number is still higher than before the law.

© 2011 Elsevier B.V. All rights reserved.

ntroduction

Tobacco smoking is a serious health concern not only for an indi-idual’s health but also for socioeconomic reasons. In Spain, it isstimated that around 55,000 deaths each year are attributable tomoking (Banegas et al., 2005). In 2006 the Spanish governmentntroduced legislation which prohibited smoking in enclosed pub-ic and work places (National Law Number: 28/2005; Ministry ofealth and Social Policy, 2005).

There have been a number of studies focusing on the impact of

moking-free laws in other countries (Erazo et al., 2010; Lock et al.,010). For instance, Erazo et al. (2010) analysed air nicotine con-entrations before and after the law, using linear regression models

∗ Corresponding author. Tel.: +34 963543232; fax: +34 963543922.E-mail address: guecor@uv.es (F. Guerrero).

955-3959/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.drugpo.2011.05.003

on log-transformed nicotine. Additionally, Lock et al. (2010) anal-ysed the behaviour of smokers in different ethnic groups, using alongitudinal and qualitative panel study.

The evolution of tobacco use in Spain is shown in Table 1. Thedata on smoking prevalence are from the 1995–2003 NationalHealth Survey (Ministry of Health and Social Policy, 2006) forthe over-16s. Those individuals who had smoked fewer than 100cigarettes during their life were considered to be non-smokers. A‘normal’ smoker was anyone who smoked less than 20 cigarettesdaily. An ‘excessive’ smoker was anyone who smoked more than20 cigarettes daily. Ex-smokers were anyone who had smoked inthe past.

As noted by Christakis and Fowler (2008, 2009), tobacco use can

be considered a socially transmitted habit and this led us to developan epidemic-type mathematical model (based on a system of dif-ferential equations) to study the evolution of tobacco use in Spainas a habit that may be spread by social contagion. This technique

248 F. Guerrero et al. / International Journal

Table 1Evolution of the proportion of non-smokers (N), normal smokers (F), excessivesmokers (Fc) and ex-smokers (E) for different years (Ministry of Health and SocialPolicy, 2006).

N F Fc E

1993 50.45% 20.59% 15.59% 13.37%1995 48.56% 21.46% 15.45% 14.53%1997 49.22% 20.17% 15.61% 15.00%

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2001 48.69% 18.98% 15.50% 16.83%2003 51.69% 18.20% 12.77% 17.34%

as been successfully applied to study other socially transmittedabits related to public health such as alcoholism (Santonja et al.,010) and cocaine consumption (Sánchez et al., 2010).

Taking account of data in Table 1, we predicted smoking preva-ence in the years after the Spanish smoke-free law came into forceJanuary, 2006) and compared these predictions (without smoke-ree law) with the real smoking rates (with smoke-free law). Thisllowed us to quantify the effect of the law by analysing the gapetween the predicted and the real prevalence rates.

ethods

he model

As noted above, we defined smoking as a disease that spreadshrough peer pressure or social contact (Christakis & Fowler, 2008,009) and proposed an epidemiological mathematical model totudy the evolution of the prevalence of smoking in the Spanishopulation (16–65 years old).

The National Health Survey for Spain (Ministry of Health andocial Policy, 2006) (16–65 years old) is divided into four sub-opulations: N, non-smokers; F, normal smokers; Fc, excessivemokers; E, ex-smokers. The transitions between these sub-opulations are shown graphically in Fig. 1.

The parameters of the model are defined as follows:

�: birth rate in Spain.0: death rate in Spain.

dF: increased death rate due to tobacco use.ˇ: transmission rate due to social pressure to adopt smoking habit.

�: rate at which an ex-smoker returns to smoking.˛: rate at which an excessive smoker becomes a normal smoker

by reducing the number of cigarettes per day.

ig. 1. Flow diagram of the mathematical model for the dynamics of smoking preva-ence in Spain. The boxes represent the sub-populations and the arrows representhe transitions between the subpopulations. Arrows are labelled by their corre-ponding model parameter.

of Drug Policy 22 (2011) 247– 251

�: rate at which a normal smoker becomes an excessive smokerby increasing the number of cigarettes per day.

�: rate at which normal smokers stop smoking.ı: rate at which excessive smokers stop smoking.

The graph shown in Fig. 1 represents a mathematical modeldescribed by a system of ordinary differential equations whosesolution gives us the evolution of the four sub-populations. Thedetails of the model building are explained in Appendix A. Sincedata in Table 1 are percentages of the total population, we had toscale the equations into the same units as the data in order to fit themodel (see Appendix A). After scaling, we obtained the followingsystem:

dn

dt= � − (d0 + �)n + d0n2 + (dF − ˇ)n(f + fc) + d0 + dF

2ne.

df

dt= ˇn(f + fc) + �e + ˛fc − (� + � + � + dF )f + d0nf + dF f (f + fc) + d0 + dF

2fe.

dfcdt

= �f − (� + + ı + dF )fc + d0nfc + dF fc(f + fc) + d0 + dF

2fce.

de

dt= �f + ıfc −

(� + � + d0 + dF

2

)e + d0ne + dF e(f + fc) + d0 + dF

2e2.

where the new scaled variables are:

n = N

P, f = F

P, fc = Fc

P, e = E

P

whilst P denotes the total population.

Parameter estimations

In order to obtain an accurate description of the evolution of thedifferent sub-populations, we had to estimate the parameters:

� = 0.01 years−1, birth rate in Spain. This takes into account theaverage Spanish rate between years 1993 and 2003 (SpanishStatistic Institute, 2010).d0 = 0.0087 years−1, is the average of the Spanish death ratebetween years 1993 and 2003 (Spanish Statistic Institute, 2010).dF = 0.0132 years−1, is the augmented death rate due to tobaccouse (Montes et al., 2004).� = 0.0425 years−1, is the rate at which ex-smokers return to smok-ing (Agudo et al., 2004).

= 0.1244 years−1, is the rate at which an excessive smokerbecomes a normal smoker by decreasing the number of cigarettesper day. We take the average rate for 1993–2003 (Ministry ofHealth and Social Policy, 2006).� = 0.1175 years−1, is the rate at which normal smokers becomeexcessive smokers by increasing the number of cigarettes per day.We take the average rate for 1993–2003 (Ministry of Health andSocial Policy, 2006).� = 0.0498 years−1, is the rate at which normal smokers stop smok-ing. We take the average rate for 1993–2003 (Ministry of Healthand Social Policy, 2006).ı = 0.0498 years−1, is the rate at which excessive smokers stopsmoking. We take the average rate for 1993–2003 (Ministry ofHealth and Social Policy, 2006).

Additionally, taking into account smoking prevalence in Spain(Table 1), we have estimated the parameter by fitting data withthe model in the mean square sense. To find the value of thatminimises the mean square error, we used the Nelder–Mead algo-rithm (Nelder & Mead, 1964; Press et al., 1986) and we obtained

= 0.0381. To see more details about the estimation of see

Santonja et al. (2010). Finally, we solved the system of differen-tial equations using NDSolve[ ] command of Mathematica (WolframResearch, 2010). Fig. 2 shows the results of the model compared toreal data from Table 1.

F. Guerrero et al. / International Journal of Drug Policy 22 (2011) 247– 251 249

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the difference was significant (Df/MSE = −1.910). The percentage ofex-smokers increased in 2006. In this case, the real data (0.2054)was greater than the predicted value (0.1773) and this difference

Table 2Real data vs. Predicted values for 2006 and 2009. MSE is the mean square error ofthe model, presented above. NA: non-available data in Spanish Statistic Institute(2009).

Real data Predicted value Difference (Df) Df/MSE

2006Non-smokers 0.4997 0.5041 −0.0044 −0.494Normal smokers 0.1856 0.1902 −0.0066 −0.742Excessive smokers 0.1094 0.1264 −0.0170 −1.910Ex-smokers 0.2053 0.1773 0.0280 3.146All smokers 0.2950 0.3186 −0.0236 −2.652

2009Non-smokers 0.4835 0.5049 −0.0214 −2.404

Fig. 2. Numerical simulation of the fitted mathematical model where the poi

esults

Our objective was to quantify the impact of the Spanish smoke-ree law introduced in 2006. We have performed simulationsith the mathematical model presented above for the period

993–2009. We have shown that our model is valid to predict thevolution of the sub-populations without the effect of the law. Ourt has a mean square error of 0.0089, or as a percentage, 0.89%.

Our quantification analysis compared the model predictions for006 and 2009 (without the effect of the law) to real data (with theffect of the law) observed for 2006 (Ministry of Health and Socialolicy, 2006) and 2009 (Spanish Statistic Institute, 2009). The realata for 2006 were collected at the end of 2006. If the impact ofhe law was significant then we would expect to see a change inhe evolution of sub-populations before and after the law. To seehether this was the case we extrapolated the model until 2009

o see what would have happened without the law and then weompared it with the real situation as recorded by the data for 2006nd 2009.

Table 2 shows the comparison between predicted and actualata for 2006 and 2009.

The first column shows the four sub-populations included inhe model and the total proportion of smokers. This last group isncluded because in the Spanish Statistic Institute (2009) the groupf smokers is not sub-divided into normal and excessive smok-rs. The second and third columns show real and predicted values,espectively. The fourth column calculates the difference betweeneal and predicted values, or in other words, the deviation withespect to the model. And the fourth column shows whether theseeviations were significant.

Our model does not fit the real data exactly (Fig. 2). There is

n average deviation of 0.0089 per point. Therefore we comparedhe deviations in Table 2, with the average deviation in order toistinguish whether deviations for 2006 and 2009 are attributableo the model or to the impact of the law. For example, a value of 3 for

resent the real data from Table 1 and the lines are the solution of the model.

the quotient Df/MSE means that the deviation between predictedand real data for that observation is three times larger than thenormal model deviation and this suggests that there is an effectonly attributable to the impact of the smoking-free law. Accordingto this, we considered significant the deviations with a value ofDf/MSE around 2 or more in absolute value.

Taking into account the Df/MSE ratios, we observed (Table 2)that the normal smoker and excessive smoker populationsdecreased in 2006. The real data (0.2950) was smaller than the pre-dicted value (0.3186) and this difference was significant (−2.652).However, this trend disappeared in 2009. In this case, the real datawas around the predicted value (without the effect of the law). Thedifference was 0.0001, much smaller than the mean square error.The excessive smoker population decreased in 2006. The observedvalue (0.1094) was smaller than the predicted value (0.1264) and

Normal smokers NA 0.1906 NA NAExcessive smokers NA 0.1240 NA NAEx-smokers 0.2017 0.1805 0.0212 2.382All smokers 0.3147 0.3146 0.0001 0.011

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50 F. Guerrero et al. / International Jo

as very significant (Df/MSE = 3.146). Unfortunately, this differ-nce did not grow over time. We noted that the real data for009 (0.2017) was again greater than the predicted value (0.1805)nd the difference was still significant (Df/MSE = 2.382), but it wasmaller than before. Finally, we saw a significant difference in non-mokers in 2009. The observed value (0.4835) was smaller thanhe predicted value (0.5049), and the difference was significantDf/MSE = −2.404). It is remarkable because this did not happen in006 when the difference was not relevant.

In summary, we can consider the initial effect of the law as con-isting of a significant decrease in the number of smokers (bothormal and excessive) and a corresponding increase in the numberf ex-smokers. The differences between the real data and predictedalues were quite significant. We can say that the global effect ofhe law in 2006 was around 2% of the total population between6 and 65 years old. We see this 2% difference between predictednd the real number of smokers (0.2950 vs. 0.3186) and betweenredicted and the real number of ex-smokers (0.2053 vs. 0.1773).owever, this trend stopped in 2009. In the survey for 2009, the dif-

erence between the predicted and the real number of ex-smokersas maintained around 2% (0.2017 vs. 0.1805), but the number of

mokers coincides with the predicted value. It is also very signifi-ant that the proportion of ex-smokers was the same in 2006 and009. Therefore, we can say that the effect of the law disappeared

n 2009. The results are in accordance with those presented in othertudies (Galan & Lopez, 2009; Ministry of Health and Social Policy,009).

iscussion

Mathematical models can be useful tools in understanding thepread of smoking, providing an insight into this problem anduantifying the impact of public health strategies. According to theesults shown above, we can conclude that when the law came intoorce in 2006 many people decided to give up smoking, decreasinghe number of smokers by 2.3% and correspondingly, increasing theumber of ex-smokers by 2.3%. Three years later, in 2009, most ofhe people who had succeeded in giving up smoking did not sufferelapse. This means that about 770,000 people stopped smoking as

result of the law (that is, the 2% of the total population between6 and 65 years old). However, 3 years later the law had no effectn new smokers, as we can see in the decreasing number of non-mokers and the increasing number of smokers reported in 2009.n order to maintain reductions in the number of new smokers, theaw should be backed up by other more specific policies.

ppendix A.

The model shown in Fig. 1 is described by a system of ordinaryifferential equations whose solution gives us the evolution overime for the four sub-populations. In order to build the model, weeed to make some assumptions:

For this model, transition time-constant parameters are moresuitable since we wanted to model the evolution of smoking habitover a short time period.

The sub-population sizes and their behaviour over time willdetermine the dynamic evolution of the smoking habit.

Complete homogeneous mixing in which every person in asub-population can interact with people from all other groups

(Murray, 2002).

The transitions between the sub-populations F(t), Fc(t) and E(t)are governed by terms proportional to the sizes of these sub-populations.

of Drug Policy 22 (2011) 247– 251

- However, the transition from N(t) to F(t) occurs through the trans-mission of the smoking habit from smokers to non-smokers,depending on the encounters amongst them. This transit is mod-elled by the term N(t)[F(t) + Fc(t)]/P(t).

- We assume that when ex-smokers relapse they do so gradually,so they enter the sub-population F(t). Hence, we do not considerthe migration between E(t) and Fc(t) in our model.

The detailed system of equations is the following (t, time inyears):

N′(t) = �P(t) − d0N(t) − ˇN(t)[F(f ) + Fc(t)]

P(t)(1)

F ′(t) = ˇN(t)[F(t) + Fc(t)]

P(t)+ �E(t) + ˛Fc(t) − (� + � + dF )F(t) (2)

F ′c(t) = �F(t) − ( + ı + dF )Fc(t) (3)

E′(t) = �F(t) + ıFc(t) −(

� + d0 + dF

2

)E(t) (4)

P(t) = N(t) + F(t) + Fc(t) + E(t) (5)

Data obtained in Table 1 are related to the percentages of pop-ulation, but the equations of the model shown above are relatedto the number of individuals. This leads us to transform (by scal-ing) the model into the same units as data, in order to comparepredictions with data.

Hence, following ideas developed in Martcheva and Castillo-Chávez (2003) and Mena-Lorca and Hethcote (1992) about how toscale models where the population is varying in size, we obtain theequations of the model as follows. Adding equations of the modelabove one gets:

P ′(t) = �P(t) − d0N(t) − dF [F(t) + Fc(t)] − d0 + dF

2E(t) (6)

Dividing both members by P(t) we have that:

P ′(t)P(t)

= � − d0N(t)P(t)

− dF

[F(t)P(t)

+ Fc(t)P(t)

− d0 + dF

2E(t)P(t)

](7)

If we define the following rates (depending on time)

n = N

P, f = F

p, fc = Fc

P, e = E

P(8)

Eq. (7) can be transformed into

P ′

P= � − d0n − dF (f + fc) − d0 + dF

2e (9)

Then, we compute the derivatives of n, f, fc and e defined in (8) using(1)–(5), (8) and (9). We obtain:

dn

dt= � − (d0 + �)n + d0n2 + (dF − ˇ)n(f + fc) + d0 + dF

2ne

df

dt= ˇn(f + fc) + �e + ˛fc − (� + � + � + dF )f + d0nf + dF f (f + fc) + d0 + dF

2fe

dfcdt

= �f − (� + + ı + dF )fc + d0nfc + dF fc(f + fc) + d0 + dF

2fce

de

dt= �f + ıfc −

(� + � + d0 + dF

2

)e + d0ne + dF e(f + fc) + d0 + dF

2e2

The solution of this scaled model allows us to obtain the predic-tion that the model gives for the evolution of the sub-populationsof non-smokers, smokers (normal and excessive) and ex-smokers.Since they are in the same units as data, we can compare themdirectly.

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