The Evolution of Age-Specific Smoking Cessation Rates in the United States From 2009 to 2018

Objective Tracking the US smoking cessation rate over time is of great interest to tobacco control researchers and policymakers since smoking cessation behaviors have a major effect on the public’s health. A couple of recent studies have employed dynamic models to estimate the US cessation rate through observed smoking prevalence. However, none of those studies has provided recent annual estimates of the cessation rate by age group. Methods We employed a Kalman filter approach to investigate the annual evolution of age-group-specific cessation rates, unknown parameters of a mathematical model of smoking prevalence, during the 2009–2018 period using data from the National Health Interview Survey. We focused on cessation rates in the 24–44, 45–64 and 65 + age groups. Results The findings show that cessation rates follow a consistent u-shaped curve over time with respect to age (i.e., higher among the 25–44 and 65 + age groups, and lower among 45–64-year-olds). Over the course of the study, the cessation rates in the 25–44 and 65 + age groups remained nearly unchanged around 4.5% and 5.6%, respectively. However, the rate in the 45–64 age group exhibited a substantial increase of 70%, from 2.5% in 2009 to 4.2% in 2017. The estimated cessation rates in all three age groups tended to converge to the weighted average cessation rate over time. Conclusions The Kalman filter approach offers a real-time estimation of cessation rates that would be helpful for monitoring smoking cessation behavior, of interest in general but also for tobacco control policymakers.


Introduction
Adult smoking prevalence in the United States decreased markedly from 2009 (20.6%) to 2018 (13.7%), according to the National Health Interview Survey (NHIS) [1]. This decreasing trend re ects desirable changes in smoking initiation and cessation rates as the result of the combined efforts of past and current tobacco control measures. While the effect of smoking initiation on mortality cannot be observed for several decades, changes in the smoking cessation rate have a near-term impact on the public's health. Thus, monitoring uctuations in the smoking cessation rate is essential for gaining insights into the changing tobacco use landscape.
Previous studies have estimated the US smoking cessation rate using different surveys and approaches [1][2][3][4][5]. In 1998, Mendez et al. used a discrete population dynamics model and NHIS prevalence data to estimate the smoking cessation rate (net of relapses) by age group [2]. Based on 1970-1993 NHIS data, the authors reported that the smoking cessation rate over both 1970-1980 and 1981-1993 increased with age. In 2022, Mendez and colleagues [1] employed another model in conjunction with 1990-2018 NHIS smoking prevalence data to quantify the average smoking cessation rate, again net of relapses, for each 6-year period from 1990 to 2019. The authors found that the smoking cessation rate increased monotonically over the entire period (2.4% during 1990-1995, 3.4% during 1997-2001, 3.5% during [2002][2003][2004][2005][2006][2007]4.2% during 2008-2013 and 5.4% during 2014-2019). In addition, they also reported an almost identical upward trend in the smoking cessation rate when using the 2002-2014 National Survey on Drug Use and Health (NSDUH) data.
In a 2012 article, Zhu et al. [4] analyzed NHIS data and found no upward trend in the estimated annual smoking cessation rate in the 1991-2010 period. Similarly, Zhuang et al. [5] reported no signi cant change in the cessation rate over time in both the 1990-2010 NHIS and 1991-2011 Tobacco Use Supplement to the Current Population Survey (TUS-CPS) data. The annual cessation rate in Zhu and Zhuang's studies [4,5] was de ned as the percentage of smokers (having smoked at least 100 cigarettes in their lifetime) who quit smoking for at least 3 months in the past year. More recently, analyses based on the 2018-2019 TUS-CPS [6], 2015 and 2017 NHIS [7,8] data show that the prevalence of recent successful cessation (quitting smoking for 6 months or longer within the past year at the time of the survey interview) generally decreased as age increased. None of these studies evaluated relapse, known to occur in a signi cant proportion of people even after having quit for six months to two years [9]. In the present work, we focused on quantifying the cessation rate as the proportion of smokers who quit each year, net of relapses (i.e., the annual net cessation rate), as in [1][2][3].
Kalman lters have the ability to generate e cient estimates of hidden states of a dynamic system based on noisy and indirect measurements from single or multiple data sources [10]. Furthermore, this technique can produce these estimates in real-time, allowing for the prompt generation of new estimates as soon as new observations become available. Kalman lters for parameter and/or state estimation have been used extensively in engineering applications [11][12][13] and other elds [14][15][16]. Attempts have also been made to employ these techniques to study infectious diseases [17][18][19]. However, no study to date has applied Kalman lters to address research questions in tobacco regulatory science where realtime surveillance of smoking behaviors is of importance and multiple data sources of the same information are available. The recent study by Mendez et. al. [1] estimated smoking cessation rate based on 1990-2018 NHIS data. However, that analysis produced the average smoking rate for each 6-year period for the entire US population (not by age groups). Tracking annual cessation rates across different age groups would be bene cial in understanding the evolution of smoking behaviors.
As such, here we use a Kalman lter-based approach to estimate annual cessation rates among adults, unknown parameters of a discrete mathematical model of smoking prevalence, by age groups (25-44, 45-64 and 65 + years old) over 2009-2018, using annual NHIS smoking prevalence data. Since smoking prevalence among 18-24-year-old smokers depends on both the smoking initiation and cessation rates for that age group, we chose to exclude it from our analysis. We stopped the analysis in 2018 due to survey design changes in 2019 that rendered smoking prevalence not directly comparable before and after that year. This is the rst study to employ a Kalman lter to estimate annual age-group-speci c smoking cessation rates. Our ndings show how age-group-speci c smoking cessation rates changed absolutely and relatively during the years studied. This approach provides tobacco regulators and researchers with a new tool to monitor changes in tobacco use trends in a timely manner.

Methods
In this section, we rst introduce a discrete mathematical model of smoking prevalence utilized to quantify annual smoking cessation rates for the three speci ed age groups (25-44, 45-64, 65+) [2].
Then, we discuss how to apply a Kalman Filter to the proposed model to estimate the unknown cessation rates using the 2009-2018 NHIS prevalence data. Here the cessation rate refers to the annual net cessation rate de ned in Mendez's work [3,20].

Mathematical model
To estimate the cessation rates by age group for each year from 2009 to 2018, we employed the following dynamic mathematical model adapted from [2].
Where and are, respectively, the number of current smokers, the death rate of current smokers, the smoking cessation rate and the US population at age a and in year . is the smoking prevalence among year-olds in year . For our three age groups, . The parameter stands for the smoking initiation rate in year . Here, we aimed at quantifying the annual net cessation rates for each of the three speci ed age groups. As such, the cessation rate was kept constant within each group which results in three unknown parameters to be estimated every year, A Kalman lter can be applied to estimate the unknown cessation rate in Expressions (1-4) by rewriting the model as follows where is the parameter noise at age in year , and are the NHIS-observed smoking prevalence and the random noise term among year-olds in year . The random term , the measurement error, was assumed to be Gaussian zero-mean white noise with the estimated standard error of the NHIS observed smoking prevalence as the standard deviation. The parameter noise was assumed to be normally distributed with zero mean. As the cessation rates are likely to be time-varying, the variance matrix of was chosen based on the "forgetting-factor" method, see Chap. 7 in [21]. Eq. (5) was used to obtain a prior estimate of the cessation rate, which was then updated using new available observation(s).
The age-speci c population for each year during the 2009-2018 period was extracted from the US Census Bureau [22]. The 2009 smoking prevalence by single age and the annual smoking prevalence by age group over the 2009-2018 period were taken from 2009-2018 NHIS [23][24][25][26]. The adult smoking initiation rate was set equal to the 25-year-old smoking prevalence in year estimated from the NHIS data. With this choice of , we captured all current smokers who initiated before age 26, but ignored the very few smokers who initiated after age 25 [27]. The effect of this omission on our results is negligible, due to the small number of individuals who start smoking after the age of 25. We used the Cancer Intervention and Surveillance Modeling Network (CISNET) age-speci c mortality rates of current smokers for each year during the considered period [28,29]. Every year the number of current smokers aged 25 is a product of the corresponding initiation rate and the 25-year-old population. The number of smokers aged 26 and over was computed as the number of smokers who were a year younger in the previous year, and who survived and continued smoking in the current year. Finally, the smoking prevalence in each of the three age groups was obtained by dividing total current smokers by the total adult population within that age group.
The weighted average cessation rate for the population aged 25 and over in year was calculated as the sum of the products of the number of current smokers in each age group at the beginning of year and the corresponding cessation rate divided by the total current smokers aged 25  Kalman lter-based methods utilize recursive Bayesian updates to estimate an unknown variable in a dynamical system while accounting for the uncertainty of observations [10]. These approaches generate a prior estimate of the unknown variable based on all the previously available observations and the given system dynamics, and then re ne this estimate with new observation(s) to obtain a posterior estimate as illustrated in Fig. 1.
To maintain a desirable balance between accuracy and computational tractability of a ltering method, we utilized the Central Difference Kalman lter (CDKF) [30,31] to estimate the annual smoking cessation rates , using the mathematical model in Equations (5-8) and the annually noisy NHIS-observed smoking prevalence. In this work, we assumed that all random variables are Gaussian [21]. The detailed algorithm of the CDKF can be found in [21,31]. The initiation rate, the NHIS-observed smoking prevalence and its estimated standard error for each age group are shown in Table A1 in the Appendix.

Results
The estimated cessation rates with their standard deviations for the three age groups, as well as the weighted average rate with its standard deviation from 2009 to 2017, are shown in Table 1. Overall, the cessation rates in the 25-44 and 65 + age groups remained nearly unchanged at approximately 4.5% and 5.6%, respectively, during the studied period. Meanwhile, the rate in the middle age group increased substantially from 2.5% in 2009 to 4.2% in 2017. The cessation rates in the youngest and oldest age groups are initially, in 2010, more than double the rate in the middle age group. However, this gap among these age groups had narrowed by 2017 with the rate among 45-64-year-olds approaching the one in the 25-44 age group. Lastly, the weighted average rate, which was computed by averaging the estimated rates of these age groups weighted by their age-group-speci c numbers of current smokers, was stable at around 3.7% until 2013 before increasing    Figure 2 displays the simulated smoking prevalence for each age group, which was obtained through the model simulations using the estimated cessation rates. The NHIS-reported smoking prevalence with its 95% CIs is also included in the gure for comparison. In all age groups, the simulated smoking prevalence aligns closely with the NHIS-observed prevalence of adult current cigarette smoking (most within the 95% con dence intervals of the NHIS-observed smoking prevalence). This lends con dence in our estimated smoking cessation rates. Figure 3 displays age-group-speci c cessation rates relative to the weighted (by the proportion of smokers in each group) average rate. During the period, we found that the magnitude of smoking cessation rates (net of relapses) is ordered in a consistent U-shape with respect to age. The cessation rate was highest among the 65 + age group, while the lowest rate was among 45-64-year-olds. However, the rates of these groups tended to converge to the weighted average value by the end of the period.
Readers can nd all the numeric values used to produce Fig. 2 in Table 1A in the Appendix.

Discussion
We have demonstrated the use of a Kalman lter approach to estimate annual smoking cessation rates, unknown parameters of a mathematical model of smoking prevalence, by age group from 2009 through 2017. Our study represents the rst of its kind to employ this methodology in tobacco control. This approach can be used to develop a monitoring tool that provides tobacco regulators and researchers with timely information about changes in tobacco use trends.
The results show that cessation rates follow a u-shaped curve with respect to age. In other words, the rate is higher among the 25-44 and 65 + age groups than among 45-64-year-olds. This is consistent with [32], where the results show that, for the period 2001-2010, the cessation rate achieved the highest value of 7.1% among 25-44-year-olds, decreasing to 4.7% for the 45-64 age group, and rising again to 5.3% for the older (65+) group. However, our results do not match with other studies [7,8] in which the authors found that the cessation rate decreased monotonically as age increased. The inconsistent ndings could be attributed to the varying de nitions of cessation rate used across different studies. For instance, the authors in [7,8] de ned this rate as the ratio of former smokers who stopped smoking for at least 6 months during the past year to the number of current smokers who smoked for more than 2 years and former smokers who stopped smoking during the past year. With this de nition, as discussed in [7], the estimated recent cessation rate is likely to overestimate the net quitting rate due to relapse by some former smokers. Meanwhile, in our present study, we estimated annual net cessation rate as in [1,3]. When comparing the cessation rate of each age group to the weighted average rate, we observed that all the estimated rates tended to start converging to the average since 2013. The weighted average cessation rate increase observed between 2014 and 2017 can be attributed to several factors. In particular, the cessation rates in the middle age group rose considerably during this period, while the rates in the other age groups were reasonably unchanged, although they remained consistently the highest age-group-speci c rates. Noticeably, the rate among 45-64-year-olds rose by 70% during the studied period, increasing from 2.5% in 2009 to 4.2% in 2017.
Our study estimated lower average smoking cessation rates during the 2009-2017 period compared with those reported in [1]. Our estimated rate averaged approximately 4.4% over 2014-2017, whereas Mendez's analysis [1] resulted in an average rate of 5.4%. It is important to note that the weighted average cessation rate reported in this study and the average rate in [1] were estimated in different populations.
Speci cally, we calculated the average cessation rate for smokers aged 25 or older, while Mendez and colleagues did for entire adult smoking population, including the 18-24 age group. The inclusion of 18-24-year-old smokers, who have been shown to have a high cessation rate [7,8], may have a signi cant impact on the reported average cessation rate in [1]. When using this Kalman lter approach to estimate the cessation rate for the entire adult smoking population (18 years of age and older), we obtained a cessation rate of 4.9%, close to that in [1].
In 2022, Mendez et. al. [1] predicted that US smoking prevalence in 2030 would be around 8.3% (95% CI: 4.6% − 16.8%). In the current work, with the 2017 input data and the estimated age-group-speci c cessation rates, we project smoking prevalence to be 8.8% (95% CI: 7.9% − 9.6%) in 2030. However, the smoking initiation rate has recently kept dropping and cessation rates, especially in the 45-64 age group, could increase beyond the period studied. As such, our ndings suggest that, while the CDC Healthy People goal of 5% (or 6.1%. revised in 2021 due to the 2019 NHIS redesign) for adult smoking prevalence in 2030 [33] is not likely to be achieved, actual prevalence could get quite close to that target.
This work has some limitations. First, the initiation rate was chosen based on the smoking prevalence of 25-year-olds, meaning that the contribution of people who started smoking after the age of 25 was not  Simulated and NHIS-observed prevalence with 95% error bars for all age groups from 2010 through 2018.