The importance of peer imitation on smoking initiation over time: a dynamical systems approach

Abstract

A recent Institute of Medicine Report calls for explicit modeling of smoking initiation, cessation and addiction processes. We introduce a model of smoking initiation that explicitly teases out the percentage of initiation due to social pressures, which we call “peer-imitation,” and the percentage due to other factors, such as media ads, family smoking, and psychological factors, which we call “self-initiation.” We propose a dynamic non-linear behavioral contagion model of smoking initiation and employ data from the National Survey on Drug Use and Health to estimate the relative contributions of imitation and self-initiation to the overall smoking initiation process. Although the percent of total smoking due to peer imitation has been trending downward over time, it remains higher than the percent due to self-initiation. We note unexpected changes for the 2007 cohort, and we discuss possible implications for intervention and for the spread of e-cigarettes.

Appendices

Appendix 1

1.1 Quitting rates

Quitting smoking is a bit more nebulous than smoking initiation. Many teen smokers quit for some period, then resume again and quit again and so on. We have therefore concentrated on initiation rates, but our model does give some information about teenage quitters. Quitting rates μ serve as average measures of the likelihood with which experimenters quit smoking (either permanently or temporarily) as they transition into more stable smoking patterns while approaching adulthood.

Our model’s estimates of quitting rates μ, for the general high school population are presented in Fig. 9. Note that the cessation rate has roughly doubled from its 0.04 value for cohort 2001–12 to its 0.08 value for cohort 2013–12.

Fig. 9

Smoking cessation rate by cohort

Appendix 2

1.1 Construction of the dynamic model

We build a mathematical model that captures the essence of the dynamics of smoking initiation and cessation. The variables and parameters of our model are carefully defined in the text. Here we give the details of the model’s construction.

We follow the Reed-Frost [12] methodology in the following way: In a period Δt, the probability that a given non-smoker takes up smoking without being influenced by his/her peers is α Δt and the probability that a non-smoker is driven by a specific smoker to initiate smoking during an encounter is κ. Then, the probability p that a non-smoker becomes a smoker in Δt is:

$$p = 1 - (1 - \alpha \cdot \Delta t) \cdot \left( {1 - \kappa } \right)^{{c \cdot \frac{S(t)}{T} \cdot \Delta t}}$$

where $c \cdot \frac{S(t)}{T} \cdot \Delta t$ is the number of meaningful contacts with a smoker in $\Delta t$.

Then, in Δt, the expected number of non-smokers who become smokers is

$$N(t) \cdot p = N(t) \cdot \left[ {1 - \left( {1 - \alpha \cdot \Delta t} \right)\left( {1 - \kappa } \right)^{{c\frac{S(t)}{T}\Delta t}} } \right]$$

(3)

Since $(1\, - \,\kappa )^{x} \, = \,e^{x\ln (1 - \kappa )} \, \approx \,1 + \left[ {\ln \,(1\, - \,\kappa )} \right]x + \frac{1}{2}\left[ {\ln \,(1\, - \,\kappa )} \right]^{2} x^{2}$,

$$(1 - \kappa )^{{\left( {c \cdot \frac{S(t)}{T} \cdot \Delta t} \right)}} \approx 1 + \left[ {ln(1 - \kappa )} \right]\left( {c \cdot \frac{S(t)}{T} \cdot \Delta t} \right) + \frac{1}{2}\left[ {ln(1 - \kappa )} \right]^{2} \left( {c \cdot \frac{S(t)}{T} \cdot \Delta t} \right)^{2}$$

Expression (3) can be approximated as:

$N(t) \cdot p \approx N(t) \cdot \left[ {1 - \left( {1 - \alpha \cdot \Delta t} \right)} \right] \cdot \left( {1 + \left[ {\ln (1 - \kappa ) \cdot c \cdot \frac{S(t)}{T} \cdot \Delta t} \right]} \right)$ + higher order terms in $\Delta t.$

$$\begin{gathered} = N(t) \cdot \left( {\beta \cdot \frac{S(t)}{T} \cdot \Delta t + \alpha \cdot \Delta t} \right) + {\text{ higher order terms in }}\Delta t \hfill \\ \approx \beta \cdot N(t) \cdot \frac{S(t)}{T} \cdot \Delta t + \alpha \cdot N(t) \cdot \Delta t. \hfill \\ \end{gathered}$$

Here, $\beta \, \equiv \, - \ln \,(1\, - \,\kappa ) \cdot c.$ Since $- \,\ln \,(1\, - \,\kappa )\, \cong \,\kappa$ for $\kappa$ small (as our κ is), we interpret β as $\kappa \cdot c$, the number of “meaningful” contacts an individual has per year times the probability that such a contact can transform a nonsmoker into a smoker. We call β “the effective contact rate.”

Similarly, the probability that a smoker becomes a non-smoker in Δt is μ Δt and the expected number of smokers who become non-smokers is: $S(t) \cdot \mu \cdot \Delta t$.

Let ΔS(t) be the change in the expected number of smokers in Δt at time t, then

$$\begin{gathered} \Delta S(t) \approx \beta \cdot N(t) \cdot \frac{S(t)}{T} \cdot \Delta t + \alpha \cdot N(t) \cdot \Delta t - \mu \cdot S(t) \cdot \Delta t \hfill \\ {\text{and}} \hfill \\ \frac{\Delta S(t)}{T} \approx \beta \cdot \frac{N(t)}{T} \cdot \frac{S(t)}{T} \cdot \Delta t + \alpha \cdot \frac{N(t)}{T} \cdot \Delta t - \mu \cdot \frac{S(t)}{T} \cdot \Delta t \hfill \\ {\text{or}} \hfill \\ \Delta s(t) \approx \beta \cdot n(t) \cdot s(t) \cdot \Delta t + \alpha \cdot n(t) \cdot \Delta t - \mu \cdot s(t) \cdot \Delta t \hfill \\ {\text{or}} \hfill \\ \frac{\Delta s(t)}{{\Delta t}} \approx \beta \cdot n(t) \cdot s(t) + \alpha \cdot n(t) - \mu \cdot s(t). \hfill \\ \end{gathered}$$

Finally, let $\Delta t \to 0:$

$$s^{\prime}(t) \equiv \frac{ds}{{dt}} = \mathop {\lim }\limits_{\Delta t \to 0} \frac{\Delta s(t)}{{\Delta t}} = \beta \cdot n(t) \cdot s(t) + \alpha \cdot n(t) - \mu \cdot s(t).$$

(4)

Since s + n = 1, rewrite (4) as:

$$\begin{gathered} s^{\prime}(t) = \beta \cdot (1 - s(t)) \cdot s(t) + \alpha \cdot \left( {1 - s(t)} \right) - \mu \cdot s(t) \\ = \alpha + s(t) \cdot \left( {\beta - \alpha - \mu } \right) - \beta \cdot s(t)^{2} . \\ \end{gathered}$$

(5)

Expand (4), using s = i + a, as:

$s^{\prime}(t) = i^{\prime}(t) + a^{\prime}(t) = \left[ {\beta \cdot n(t) \cdot s(t) - \mu \cdot i(t)} \right] + \left[ {\alpha \cdot n(t) - \mu \cdot a(t)} \right]$.

For peer-influenced smoking initiation,

$$i^{\prime}(t) = \beta \cdot n(t) \cdot s(t) - \mu \cdot i(t).$$

(6)

For self-initiators,

$$a^{\prime}(t) = \alpha \cdot n(t) - \mu \cdot a(t).$$

(7)

For non-smokers n(t),

$$\begin{gathered} n^{\prime}(t) = - s^{\prime}(t) = \mu \cdot s(t) - \alpha \cdot n(t) - \beta \cdot n(t) \cdot s(t) \\ = \mu \cdot s(t) - \alpha \cdot \left( {x(t) + q(t)} \right) - \beta \cdot \left( {x(t) + q(t)} \right) \cdot s(t) \\ \end{gathered}$$

(8)

Using n = x + q, expand (8) as:

$$n^{\prime}(t) = x^{\prime}(t) + q^{\prime}(t) = \left[ { - \alpha \cdot x(t) - \beta \cdot x(t) \cdot s(t)} \right] + \left[ {\mu \cdot s(t) - \alpha \cdot q(t) - \beta \cdot q(t) \cdot s(t)} \right],$$

where the dynamic of the never-smokers is:

$$x^{\prime}(t)\, = \, - \,x(t) \cdot \left[ {\alpha \, + \,\beta \cdot s(t)} \right]$$

(9)

and the dynamic of the non-smoking former smokers is:

$$q^{\prime}(t)\, = \,\mu \cdot s(t)\, - \,\alpha \cdot q(t)\, - \,\beta \cdot q(t) \cdot s(t)$$

(10)

Equations (6), (7), (9), and (10) form a system of four differential equations in i(t), a(t), x(t), and q(t). Our empirical data use numbers of smokers s(t) over time and numbers of never-smokers x(t) over time. Hence, we use Eqs. (3) and (9) as a system in s(t) and x(t) to estimate the parameters $\alpha ,\,\beta ,\,\mu .$:

$$\begin{gathered} s^{\prime}(t)\, = \,\alpha \, + \,s(t) \cdot \left( {\beta \, - \,\alpha \, - \,\mu } \right)\, - \,\beta \cdot s(t)^{2} \\ x^{\prime}(t)\, = \, - \,x(t) \cdot \left[ {\alpha + \beta \cdot s(t)} \right] \\ \end{gathered}$$

(11)