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Mathematical Biosciences
journal homepage: www.elsevier.com/locate/mbs
Adolescent vaping behaviours: Exploring the dynamics of a social contagion
model
Sarah I. Machado-Marques, Iain R. Moyles ∗
Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, M3J1P3, ON, Canada
ARTICLE INFO
MSC:
37N25
62P25
41A60
37C75
Keywords:
Social contagion
Vaping
e-cigarettes
Mathematical model
Social influence
Equilibrium
Stability
Oscillations
ABSTRACT
Vaping, or the use of electronic cigarettes (e-cigarettes), is an ongoing issue for public health. The rapid increase
in e-cigarette usage, particularly among adolescents, has often been referred to as an epidemic. Drawing upon
this epidemiological analogy between vaping and infectious diseases as a theoretical framework, we present a
deterministic compartmental model of adolescent e-cigarette smoking which accounts for social influences
on initiation, relapse, and cessation behaviours. We use results from a sensitivity analysis of the model’s
parameters on various response variables to identify key influences on system dynamics and simplify the
model into one that can be analysed more thoroughly. We identify a single feasible endemic equilibrium
for the proportion of smokers that decreases as social influence on cessation increases. Through steady state
and stability analyses, as well as simulations of the model, we conclude that social influences from and on
temporary quitters are not important in overall model dynamics, and that social influences from permanent
quitters can have a significant impact on long-term system dynamics. In particular, we show that social
influence on cessation can induce persistent recurrent smoking outbreaks.
1. Introduction
Vaping, or the use of electronic cigarettes (e-cigarettes), is an on-
going issue for public health. Introduced to the public around 2007
and marketed as a safer alternative to traditional cigarettes, vaping
rapidly gained popularity, particularly among adolescents [1,2]. An
e-cigarette is a small, often pen-sized product with a cartridge that
is heated in order to produce an odourless vapour [1]. Cartridge
solutions can contain numerous harmful substances, including high
doses of nicotine, that are inhaled deep into the lungs and serve as
pulmonary irritants [1,2]. Although the long term effects of vaping
are still unknown due to their novelty, short term effects have in-
cluded hospitalization due to e-cigarette/vaping product use-associated
lung injury (EVALI), nicotine dependence, and an increased risk for
initiating cigarette use [2–5]. The use of e-cigarettes poses potential
health risks for users of any age, however adolescents are particularly
vulnerable to becoming users and experiencing these adverse effects
due to the marketing of vaping products [6]. They use a wide variety
of available flavours and customizability to appeal to adolescents [1].
The impact of these highly effective marketing strategies include a
minimized perception of the risks of e-cigarette usage and an increased
curiosity in the behaviour [1,6–8].
∗Corresponding author.
E-mail addresses: smarques@yorku.ca (S.I. Machado-Marques), imoyles@yorku.ca (I.R. Moyles).
Over the years, the growth in adolescent-aged users has become an
alarming trend. In the United States, adolescent vaping was declared
an ‘‘epidemic’’ by the Surgeon General in 2018 [1]. According to the
2019 Canadian Health Survey of Children and Youth administered by
Statistics Canada, nearly one in four adolescents between the ages of
12 and 17 reported vaping daily or almost daily [9]. Importantly, e-
cigarettes do not have the same level of regulation as other tobacco
products, making them of particular concern for public health. For
example, in Canada, flavoured cigarettes have been banned, while there
has been no such legislation passed for e-cigarettes on the federal
level [10,11]. As such, it is important to determine the trajectory
of the number of adolescent e-cigarette users and to understand the
underlying dynamics of the usage.
Our focus is on advancing the understanding of social contagions
in mathematical models by incorporating research from the social
sciences to more accurately model how one’s social environment im-
pacts their usage of addictive substances [12]. Equipped with this type
of information, public health efforts can be better and more quickly
implemented in order to address the concern. The use of a social
modelling framework requires evidence grounding it in psychological
and sociological concepts, which can often be overlooked in smoking-
type models [13–16]. Given the epidemic analogy commonly made,
https://doi.org/10.1016/j.mbs.2024.109303
Mathematical Biosciences 377 (2024) 109303
Available online 17 September 2024
0025-5564/© 2024 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ).
S.I. Machado-Marques and I.R. Moyles
the problem is typically studied through the lens of social contagion,
which the American Psychological Association defines as ‘‘the spread
of behaviours, attitudes, and affect through crowds and other types
of social aggregates from one member to another’’ [17]. Importantly,
social contagions can occur with or without the presence of pressures
to conform or the desire to imitate [17].
It is fairly well established in the psychological literature that
adolescents are prone to risk contagion when among their peers [18].
Further, studies show that an adolescent’s close connections can have
a significant impact on their choice to begin using e-cigarettes, with
friends being the most common source of e-cigarettes and the majority
of youth users initiating while ‘‘hanging out with friends’’ [6,19].
The presence of peers who engage in vaping behaviours also serves
to normalize the use of e-cigarettes in the population by increasing
curiosity about the behaviour and increasing future initiation risk [6].
It is apparent that an adolescent’s social connections, interactions, and
environment are key factors in their decision of whether or not to begin
smoking e-cigarettes. Therefore, it is justified to view the issue through
the lens of social contagion. By doing so, new interventions targeting
peer influences on the increasing levels of vaping adolescents can begin
to be considered, as is currently being recommended by researchers in
the field [6,19].
One question which naturally arises is that of the difference between
modelling nicotine consumption versus the usage of other substances,
namely opioids, cannabis, and alcohol, in the adolescent population.
Historically, the opioid crisis has been fuelled largely by prescription
drugs and the influence of the pharmaceutical industry [20,21]. In the
case of cannabis, the prevalence of adolescent usage is lower and a
smaller proportion of these individuals become substance-dependent
with a slow transition time [21,22]. While alcohol is more widely
consumed than nicotine, people remain more casual consumers and are
less likely to become dependent [21].
Past efforts to deterministically model smoking-type behaviours
use an SIR-type disease model framework with an analogy to smok-
ing [23]. Extensions of this model account for possible social mech-
anisms involved in the process of relapse, a state of temporary quit-
ting, the effects of temporary quitters choosing to become perma-
nent quitters, and compartments representing smoking frequency (ie.
light/occasional and heavy smokers) [13–15,23–25]. Notably, these
models only consider social contagion in the processes of initiation and
relapse. To the best of our knowledge, there are no deterministic models
which explore the impact of social influences and shifting social norms
on cessation of smoking-type behaviours. Social influence on cessation
of smoking-type behaviours is not a well-studied or understood topic.
However, there is a small body of literature which suggests that so-
cial connections can influence quitting behaviours [26–28]. While this
literature is not exactly within the context of adolescent vaping, it still
indicates that social influences on cessation are worthwhile to consider.
In this paper, we formulate a new deterministic compartmental
model of adolescent e-cigarette smoking which accounts for social
influences on initiation, relapse, and cessation behaviours. We present
this model in Section 2, where we derive and analyse the stability of
a smoking-free equilibrium and look at model sensitivity. Motivated
by the sensitivity analysis, we derive a reduced model in Section 3,
where we uncover and analyse the feasibility and stability of smoking-
present equilibria. We support our analysis with numerical simulations
in Section 4. We discuss the implications of our results and offer
conclusions in Section 5.
2. Socially-influenced contagion model
To model the usage of e-cigarettes in the adolescent population, the
model structure previously presented in Sharomi and Gumel (2008),
will be used as a basis [15]. We also include the idea of a nonlinear
relapse rate, as was suggested in Alkhudari et al. (2014) [14]. By
combining and extending the models, we explore the effects of (1)
Fig. 1. Socially-influenced contagion model flow diagram.
socially-driven initiation and relapse, (2) relapse due to nicotine de-
pendency and other non-socially related factors, (3) socially-influenced
cessation, and (4) cessation of one’s own volition as they pertain to
overall system dynamics.
Similar to Sharomi and Gumel, we consider the adolescent popu-
lation to be comprised of the following four separate compartments:
potential e-cigarette smokers (𝑃), e-cigarette smokers (𝑆), temporary
quitters (𝑄𝑡), and permanent quitters (𝑄𝑝) [15]. The model assumptions
made are as follows:
•Initiation of vaping behaviours only occurs as a result of social
contagion from current e-cigarette smokers.
•All individuals age into adolescence as potential e-cigarette smok-
ers.
•The size of the adolescent population remains constant over time,
meaning adolescents age into and out of the population at the
same rate.
•The adolescent population is well-mixed, although this mixing
does not strictly require close physical proximity. For example,
it is possible to make social contact with others and potentially
influence them over the internet.
Adolescents who are potential e-cigarette smokers (𝑃) only begin
vaping due to social contagion from e-cigarette smoking peers (𝑆) at a
rate 𝛽 > 0. Once an adolescent is vaping, they quit of their own volition
at a rate 𝛾1>0, quit due to social influences from temporary quitters
(𝑄𝑡) at a rate 𝛾2>0, or quit due to social influences from permanent
quitters (𝑄𝑝) at a rate 𝛾3>0. Of those who quit, a proportion 𝜎∈ (0,1)
of them will become permanent quitters. The remaining proportion
(1 − 𝜎) will become temporary quitters. Those who become temporary
quitters relapse and become e-cigarette smokers again from nicotine
dependency and other non-social/natural factors at a rate 𝛼1>0, or
due to social contagion from other e-cigarette smokers at a rate 𝛼2>0.
Further, individuals age into and out of adolescence at a rate 𝜇 > 0. We
note that while mathematically similar, our parameter definition of 𝜇is
in contrast to death of smokers used in [15]. We note that the proposed
model restricts social influence on adolescents from within their own
peer group. While this is generally supported [29], we acknowledge
that it is a limiting assumption, particularly in that adults may play an
influential role on initiating or quitting in a familial context. As was
done in Alkhudari et al. (2014), a normalized population size is used
such that 𝑃+𝑆+𝑄𝑡+𝑄𝑝= 1, where 𝑃,𝑆,𝑄𝑡, and 𝑄𝑝≥0represent
proportions of the adolescent population [14]. The flow diagram for
the socially-influenced contagion model is illustrated in Fig. 1.
The resultant system of ODEs for the socially-influenced contagion
model is
d𝑃
d𝑡=𝜇−𝛽𝑃 𝑆 −𝜇𝑃 , (1a)
Mathematical Biosciences 377 (2024) 109303
2
S.I. Machado-Marques and I.R. Moyles
d𝑆
d𝑡=𝛽𝑃 𝑆 + (𝛼1+𝛼2𝑆)𝑄𝑡− (𝛾1+𝛾2𝑄𝑡+𝛾3𝑄𝑝+𝜇)𝑆, (1b)
d𝑄𝑡
d𝑡=(1 − 𝜎)(𝛾1+𝛾2𝑄𝑡+𝛾3𝑄𝑝)𝑆− (𝛼1+𝛼2𝑆+𝜇)𝑄𝑡,(1c)
d𝑄𝑝
d𝑡=𝜎(𝛾1+𝛾2𝑄𝑡+𝛾3𝑄𝑝)𝑆−𝜇𝑄𝑝.(1d)
The inherent assumption that adolescents can only initiate vaping
behaviours as a consequence of social contagion and interaction with
e-cigarette smokers means it is additionally required that 𝑆(0) >0in
order for there to be an initial uptake in the behaviour. The value of
𝑆(0) >0can be considered to be the result of some stochastic effect,
where all those adolescents who initiated the behaviour regardless of
social influence have already done so by 𝑡= 0.
We do not explicitly consider the influence of media on the decision
to smoke or quit. When vaping was a new technology, there were
several marketing tools suggesting that e-cigarettes were safer than tra-
ditional cigarettes [1,7]. Many companies released various e-cigarette
flavours which may have appealed to adolescent curiosity [8]. From
the perspective of our model, we assume that these early promotional
media tools would influence 𝑆(0). There have also been media cam-
paigns to discourage the use of tobacco products in youth [30]. Many
adolescents are aware of these campaigns [31] and exposure to these
campaigns may decrease susceptibility to e-cigarette initiation [32].
However, a study on Glasgow teens indicated that smoking initiation
was more present in non-smokers who had exposure to smoking peers
compared to those who had little social influence from smokers [33].
Presumably, both groups would be exposed to similar outside media in-
fluence and so we assume social-influence has a more important explicit
dependence than media. Media influence on quitting can implicitly be
captured with the parameter 𝛾1.
Since the model (1) is an adaptation of one for traditional smoking
in the general population, it is important to contextualize its usage
for adolescent vaping, particularly considering that the addictiveness
of nicotine is comparable regardless of delivery device [34]. Firstly, e-
cigarette usage in youth and young adults is much higher than in older
adults [35]. This increase in popularity could be because a significant
portion of adolescents perceive e-cigarettes to be a safer, more socially
acceptable alternative to cigarettes [7]. Secondly, most adult smokers
initiated smoking in their adolescence [36] and nearly 70% of those
initiated because of social influence [37]. Furthermore, those that
started smoking in their adolescence showed higher nicotine depen-
dence and lower cessation attempts later in adulthood [38]. Adults also
have different relationships with e-cigarettes. Dual use of traditional
cigarettes and e-cigarettes as well as using vaping as a cessation aid
for traditional cigarettes is especially common in older adults [39].
Overall, the heterogeneity in usage, reason for usage, and differing
mechanisms for cessation in adults suggest that focusing on adolescents
is appropriate. The usage of e-cigarettes is up to ten times higher as a
nicotine source compared to any other source suggesting that vaping
as the primary mechanism of smoking is also appropriate [36].
An important aspect of limiting our model to adolescents is that
sale of tobacco and nicotine products is often illegal in many jurisdic-
tions. Thus, this may be a barrier to curiosity as a mode of smoking
initiation which overall may strengthen the argument of importance
of social influence in smoking behaviour. For example, in a survey
of approximately 10 000 adolescents in the United States nearly 80%
of respondents indicated that they obtained e-cigarettes from a social
source [40].
2.1. Local stability of smoking-free equilibrium (SFE)
The model’s SFE (𝑃∗, 𝑆∗, 𝑄∗
𝑡, 𝑄∗
𝑝) = (1,0,0,0) represents a pop-
ulation of only e-cigarette non-users. In order to investigate local
stability of the SFE, the next generation matrix method is employed,
as described by van den Driessche and Watmough [41]. Accordingly,
the ‘‘smoke-present’’ compartments, 𝑆and 𝑄𝑡, are identified as the
traditional infection compartments. The differential equations for these
compartments can be written as
d𝑆
d𝑡
d𝑄𝑡
d𝑡
=−,(2)
where
=𝛽𝑃 𝑆
0(3)
is the vector of terms that produce new smokers in a compartment, and
=−(𝛼1+𝛼2𝑆)𝑄𝑡+ (𝛾1+𝛾2𝑄𝑡+𝛾3𝑄𝑝+𝜇)𝑆
−(1 − 𝜎)(𝛾1+𝛾2𝑄𝑡+𝛾3𝑄𝑝)𝑆+ (𝛼1+𝛼2𝑆+𝜇)𝑄𝑡(4)
is the vector of terms that represent the transfer of individuals in and
out of the compartments by other means. The Jacobian matrices of
and are 𝐹and 𝑉, respectively, where
𝐹=𝛽𝑃 0
0 0,(5a)
𝑉=−𝛼2𝑄𝑡+𝛾1+𝛾2𝑄𝑡+𝛾3𝑄𝑝+𝜇−(𝛼1+𝛼2𝑆) + 𝛾2𝑆
−(1 − 𝜎)(𝛾1+𝛾2𝑄𝑡+𝛾3𝑄𝑝) + 𝛼2𝑄𝑡−(1 − 𝜎)𝛾2𝑆+𝛼1+𝛼2𝑆+𝜇.(5b)
The Jacobian matrices evaluated at the SFE are
𝐹𝑆𝐹 𝐸 =𝛽0
0 0, 𝑉 𝑆 𝐹 𝐸 =𝛾1+𝜇−𝛼1
−(1 − 𝜎)𝛾1𝛼1+𝜇(6)
with spectral radius 𝜌(𝐹 𝑉 −1 ) = 0, where
0=𝛽
(𝛾1+𝜇)(1 − 𝜓), 𝜓 =𝛼1
𝛼1+𝜇
⋅
𝛾1
𝛾1+𝜇(1 − 𝜎).(7)
Thus, the SFE of the socially-influenced contagion model is locally
asymptotically stable if 0<1, but unstable if 0>1. In the context
of disease outbreaks, the basic reproduction number 0is typically
defined as the average number of infections produced by a single
infectious individual in a fully susceptible population during their
entire infectious period [42]. In this context, the definition is adapted
such that 0can be interpreted as the expected number of secondary e-
cigarette users arising from a single e-cigarette user during their entire
smoking period in a population of only e-cigarette non-users.
We note that the social influences on relapse (𝛼2) and cessation (𝛾2,
𝛾3) do not appear in (7), indicating they are not influences in the initial
growth phase of the spread of vaping behaviours. As these behaviours
were the major model modification from the model of Sharomi and
Gumel [15], the basic reproduction number of our model is the same
as theirs.
The expression (7) for 0can be interpreted sociologically. The
𝛽∕(𝛾1+𝜇)term represents the spread of behaviours from one vaping
individual into a fully non-vaping population over the course of their
smoking period when relapse is ignored. However, the relapse function
(1 − 𝜓) ∈ (0,1), serves to increase 0due to the relapse of temporary
quitters, as 𝜓is the proportion of smokers who quit temporarily and
revert back to smoking. Ultimately, increasing 𝜓would increase one’s
average smoking period, and thus increases 0.
2.2. Sensitivity analysis
The behavioural role of relapse and cessation do not influence the
onset of smoking as demonstrated by (7). To explore the influence of
this behaviour on the entirety of the smoking dynamic, we conduct a
sensitivity analysis to gain preliminary insight into which factors play
the most important role in model interpretations. A sensitivity analysis
is a quantitative approach that can provide important information
on the relationship between model parameters and outcomes. It also
describes how uncertainty in parameters may impact model dynamics
given the lack of estimates in e-cigarette literature.
Mathematical Biosciences 377 (2024) 109303
3
S.I. Machado-Marques and I.R. Moyles
Table 1
Sensitivity analysis parameter assumptions.
Parameter Interpretation Value or range of
uniform PDF
𝜇Rate of removal in social network 1/8 years−1
𝛽Rate of smoking initiation [1, 4] years−1
𝛼1Rate of natural relapse in temporary
quitters
[18, 45] years−1
𝛼2Rate of socially-influenced relapse in
temporary quitter
[18, 45] years−1
𝛾1Rate of natural quitting of smoking [1, 4] years−1
𝛾2Rate of social influence from
temporary quitters on quitting
smoking
[1, 4] years−1
𝛾3Rate of social influence from
permanent quitters on quitting
smoking
[1, 4] years−1
𝜎Proportion of quitters who are
permanent
[0.01, 0.2]
The methods of Latin hypercube sampling (LHS) and partial rank
correlation coefficients (PRCCs) are used to perform our global sen-
sitivity analysis on the model [43]. For our purposes, only uniform
distributions will be sampled from, given that the probability distri-
bution functions of the parameters are unknown. A summary of the
parameter assumptions is given in Table 1. Adolescence can be defined
as the period of transition between childhood and adulthood, which is
roughly from 11 to 19 years old [44]. Therefore, the fixed value 𝜇= 1∕8
years−1is assumed. Through survival analysis, it has been found that
nicotine users who are able to quit for at least a year are unlikely to
return to using [45]. Therefore, it is assumed that if a never-smoker
(𝑃) has not been influenced after a year of contact with a smoker, then
they are unlikely to be influenced at all as they are not as predisposed
as temporary quitters. Thus, the lower bound of 𝛽is assumed to be
1 year−1. An upper bound of 4 years−1(or three months) is assumed.
Given that the relationship between initiation rates and quitting rates
is unknown, we avoid making assumptions on the relationship and
assume the ranges for 𝛾1, 𝛾2, and 𝛾3to be the same. Relapse curves
for nicotine usage show that relapse mostly occurs within days to two
weeks after the cessation attempt for regular users [45,46]. In our
model, we allow the 𝑆compartment to be made up of any type of
user, including light users, and so we shift the range and assume relapse
occurs between roughly 8 to 20 days, which gives the range of 18 to
45 years−1for 𝛼1and 𝛼2. Finally, it is assumed that the proportion
of cessation attempts which are successful (𝜎) is low [47]. An LHS
sample size of 3000 was used, with initial conditions of 𝑃(0) = 0.92
and 𝑆(0) = 0.08 [48].
The results of the sensitivity analysis can be found in Fig. 2. Pa-
rameters associated with producing smokers (𝛽, 𝛼1, 𝛼2)were found to
be positively correlated with the response variables of 0, peak value,
smokers’ equilibrium, and final size. However, the correlations of 𝛼1
and 𝛼2prove to be insignificant, indicating that relapse is not an
important factor in the metrics for the parameter ranges tested. This
seems to be due to relapse occurring quite quickly, where the residence
time of an individual in the temporary smoker compartment is so short
that they are effectively leaving the smoking compartment and re-
entering almost immediately. The results also indicate that 𝛽is the
most significant factor in the severity of e-cigarette smoking in the
population. The parameters associated with producing smokers were,
however, negatively correlated with peak time, with 𝛽being the most
contributing factor.
Parameters associated with smoking cessation (𝛾1, 𝛾2, 𝛾3, 𝜎) were
found to be negatively correlated with all response variables, with 𝜎
emerging as the most significant parameter across all response vari-
ables. As the proportion of quitters who are able to quit permanently
decreases, the more they are able to re-initiate smoking and increase
the average smoking lifetime. As a result, they are able to also ensure
that vaping becomes more widespread. The effect of temporary quitters
on e-cigarette smoking cessation, captured in 𝛾2, was found to be
insignificant overall. Again, this seems to be due to the proportion
of the population in the temporary quitter compartment remaining
relatively small in the simulations. The non-social cessation rate of
𝛾1was found to be significant in relation to 5 out of the 6 response
variables.
The most interesting finding arising from the sensitivity analysis is
the significance of 𝛾3on the smokers’ equilibrium and final size values.
The results show that 𝛾3is not influential on 0, peak value, or peak
time. This was corroborated by its absence in the basic reproduction
number (7). However, while 𝛾3is not influential early in the dynamics,
it is a significant factor in the long-term trajectory of e-cigarette usage
in the population.
Overall, parameters found to be insensitive are 𝛼1,𝛼2, and 𝛾2. Note
that 𝛼2and 𝛾2are not present in the expression for 0, so their lack
of influence is unsurprising. While 𝛼1is in the expression for 0, we
know that as 𝜇⟶0,0⟶𝛽∕𝛾1𝜎, provided 𝛼1is not very small. If 𝛼1
is very small, then the ratio 𝛼1∕(𝛼1+𝜇)will be small. This means that
1 − 𝜓will approach 1. In either case, 𝛼1will have little influence in 0.
2.3. Smoking-present equilibria (SPE)
The introduction of modified relapse and cessation behaviours adds
complexity to the long-time smoking dynamics as revealed by the
sensitivity analysis. Before proceeding with the SPE, we first non-
dimensionalize time to scale with 𝛽−1, yielding the non-dimensional
model
𝑃=𝜖(1 − 𝑃) − 𝑃 𝑆, (8a)
𝑆=𝑃 𝑆 +𝑎(1 + 𝑎𝑆)𝑄𝑡−𝑟(1 +
𝛤−1𝑄𝑡+𝛤−1 𝑄𝑝)𝑆−𝜖𝑆, (8b)
𝑄𝑡=(1 − 𝜎)𝑟(1 +
𝛤−1𝑄𝑡+𝛤−1 𝑄𝑝)𝑆−𝑎(1 + 𝑎𝑆)𝑄𝑡−𝜖 𝑄𝑡,(8c)
𝑄𝑝=𝜎𝑟(1 +
𝛤−1𝑄𝑡+𝛤−1 𝑄𝑝)𝑆−𝜖𝑄𝑝,(8d)
where the over-dot indicates differentiation with respect to the non-
dimensional time and we introduce the non-dimensional parameters
𝑟=𝛾1
𝛽, 𝑎 =𝛼1
𝛽, 𝑎 =𝛼2
𝛼1
,
𝛤=𝛾1
𝛾3
,
𝛤=𝛾1
𝛾2
, 𝜖 =𝜇
𝛽.
(9)
Generally, people change their social influence structure slower than
they are exposed to smoking pressures (see Table 1) and so we will be
able exploit that 𝜖 ≪ 1is a small parameter.
The SPE of the socially-influenced contagion model have the form
(𝑃∗∗, 𝑆 ∗∗, 𝑄∗∗
𝑡, 𝑄∗∗
𝑝), where
𝑃∗∗ =𝜖
𝑆∗∗ +𝜖,(10a)
𝑄∗∗
𝑡=𝜖𝑟𝛤 (1 − 𝜎)𝑆∗∗
(𝑎+𝜖+𝑎 𝑎𝑆 ∗∗)(𝜖𝛤 −𝑟𝜎𝑆 ∗∗) − 𝜖𝑟𝛤
𝛤−1(1 − 𝜎)𝑆∗∗ ,(10b)
𝑄∗∗
𝑝=𝜎𝑟𝛤 𝑆 ∗∗
𝜖𝛤 −𝜎𝑟𝑆 ∗∗ −𝜖𝑟𝛤
𝛤−1(1 − 𝜎)(𝑎+𝜖+𝑎 𝑎𝑆 ∗∗)−1 𝑆∗∗ ,(10c)
while 𝑆∗∗ satisfies the cubic polynomial
𝑎 𝑎𝑟𝜎𝑆 ∗∗3+𝐴𝑆∗∗2+𝐵𝑆 ∗∗ +𝜖𝛤 𝐾 (0− 1) = 0,(11)
where
𝐴=𝑟𝜎(𝑎+𝜖) − 𝑎 𝑎(𝑟𝜎 (1 + 𝛤−𝜖) + 𝜖𝛤 ) + 𝜖𝑟𝛤
𝛤−1(1 − 𝜎),(12a)
𝐵=𝜖𝛤 𝑎 𝑎(1 − 𝜖−𝜎 𝑟) − 𝜖𝑟𝛤
𝛤−1(1 − 𝜎)(1 − 𝜖)(12b)
− (𝑎+𝜖)(1 − 𝜖)𝑟𝜎 −𝐾𝛤 ,
𝐾=𝜖2+𝜖(𝑎+𝑟) + 𝑎𝑟𝜎. (12c)
When 𝛼2=𝛾2=𝛾3= 0 then we recover the model of [15] and (11)
reduces to
𝑆−𝜖(0− 1) = 0 (13)
Mathematical Biosciences 377 (2024) 109303
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S.I. Machado-Marques and I.R. Moyles
Fig. 2. PRCC analyses for the improved social contagion model using 3000 LHS samples for the parameter ranges in Table 1, where the red horizontal lines at ±0.5indicate the
threshold for significance and * denotes a statistically significant result (𝑝-value <0.05).
yielding the single SPE equilibrium from their work. The cubic roots of
(11) produce possible additional SPE for our modified model, however
closed form expressions are cumbersome. Instead, we will exploit that
𝜖 ≪ 1posing a perturbative expansion,
𝑆∗∗ ∼𝑠0+𝜖𝑠1+ … ,(14)
which after substitution into (11) produces to leading order,
𝑠0(𝑎𝑠0+ 1)(1 + 𝛤−𝑠0),(15)
furnishing the roots
𝑠(1)
0=0,(16a)
𝑠(2)
0= − 𝑎−1 ,(16b)
𝑠(3)
0=1 + 𝛤 . (16c)
The first of these roots is the leading order of the SPE given by (13)
first reported in [15] and the second root is infeasible since 𝑎 > 0. We
require 𝑆 < 1for feasibility and the third root appears to satisfy 𝑠(3)
0>1.
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S.I. Machado-Marques and I.R. Moyles
If 𝛤is small then 𝑠(3)
0∼ 1 and perturbative corrections may allow it to
be feasible. However, even in this case, returning to (10c) we would
see that 𝑄∗∗
𝑝becomes negative to leading order meaning 𝑠(3) is always
infeasible.
Therefore, 𝑠(1)
0remains the only feasible root to (11) and we de-
termine its leading order behaviour by seeking corrections to (𝜖)
yielding,
𝑠1=𝛤(1 − 𝑟𝜎)
𝑟𝜎(1 + 𝛤),(17)
so that overall there remains a single feasible smoking present equilib-
rium,
𝑠∼𝜖𝛤(1 − 𝑟𝜎)
𝑟𝜎(1 + 𝛤)∼𝜖𝛤
1 + 𝛤(0− 1).(18)
Thus, we see compared to (13) that the role of social influence from
permanent quitting reduces the final-size of the smoking population.
3. Reduced socially-influenced contagion model
The results of the sensitivity analysis only identified four parameters
as being significant: 𝛽,𝛾1,𝛾3, and 𝜎. This was further corroborated
through the SPE analysis in Section 2.3 where it was shown that 𝛾2did
not influence any of the leading order SPE and that 𝛼2was only present
at leading order in an infeasible root. We therefore consider a reduced
model where we take 𝛼2=𝛾2= 0 (equivalently 𝑎 = 0 and
𝛤→∞).
Despite not being sensitive, we retain 𝛼1in the reduced model for two
reasons. Firstly, it preserves the basic reproduction number of the SFE
in its entirety, not just to leading order, and secondly, if terms with
parameters 𝛼1and 𝛼2were all removed, then temporary quitters would
effectively become permanent quitters. Thus, the model’s structure
is sensitive to relapse. Aside from preserving the Basic Reproduction
Number, choosing 𝛼1to remain over 𝛼2assumes that relapse due to
nicotine dependence and cravings is a more prevalent factor than
socially-influenced relapse as has been noted in the literature [49]. The
resultant non-dimensional ODEs are:
𝑃=𝜖(1 − 𝑃) − 𝑃 𝑆, (19a)
𝑆=𝑃 𝑆 +𝑎𝑄𝑡−𝑟(1 + 𝛤−1𝑄𝑝)𝑆−𝜖 𝑆, (19b)
𝑄𝑡=(1 − 𝜎)𝑟(1 + 𝛤−1𝑄𝑝)𝑆− (𝑎+𝜖)𝑄𝑡,(19c)
𝑄𝑝=𝜎𝑟(1 + 𝛤−1 𝑄𝑝)𝑆−𝜖𝑄𝑝.(19d)
This reduced model carries the same assumptions as the full model,
however socially-influenced relapse and cessation influences by tempo-
rary quitters are no longer considered.
3.1. Local stability of the smoking present equilibrium
By design, the reduced model retains the SFE of (𝑃∗, 𝑆∗, 𝑄∗
𝑡, 𝑄∗
𝑝) =
(1,0,0,0) and following the next generation method maintains the same
basic reproduction number. The SPE now satisfy,
𝑃∗∗ =𝜖
𝑆∗∗ +𝜖, 𝑄∗∗
𝑡=(1 − 𝜎)𝑟𝛤 𝜖𝑆 ∗∗
(𝑎+𝜖)(𝛤 𝜖 −𝜎𝑟𝑆 ∗∗), 𝑄∗∗
𝑝=𝜎𝑟𝛤 𝑆 ∗∗
𝜖𝛤 −𝜎𝑟𝑆 ∗∗ ,
(20)
where 𝑆∗∗ now satisfies
𝑆∗∗2−𝑎1𝑆∗∗ +𝛤 𝜖𝑎0= 0,(21)
with
𝑎0=𝑎(1 − 𝑟𝜎) + 𝜖(1 − 𝑎−𝑟) − 𝜖2
𝑟𝜎(𝑎+𝜖),(22a)
𝑎1=𝑟𝜎𝑎(𝛤+ 1) − 𝜖(𝑟𝜎(𝑎− 1) − 𝛤(𝑎+𝑟)) + 𝜖2(𝑟𝜎 −𝛤)
𝑟𝜎(𝑎+𝜖).(22b)
The leading order roots of this are
𝑆∗∗ ∼𝜖𝛤(1 − 𝑟𝜎)
𝑟𝜎(1 + 𝛤), 𝑆 ∗∗ ∼ 1 + 𝛤 , (23)
matching the positive roots (16) of the full model (1). Thus, the reduced
model (19) captures the behaviour of the full model with the reduced
complexity coming at the loss of the infeasible negative root (16b). We
more thoroughly compare the full and reduced model in Appendix to
show that there is minimal difference between the two.
When social influence from quitting was not considered, Sharomi
and Gumel showed that the single smoking present equilibrium was
locally asymptotically stable [15]. However, the dynamic structure of
the model towards equilibrium was not discussed. This is an important
distinction because in the context of an evolving social contagion,
intermediary dynamics which may differ from a long-term steady state
will influence public policy. Exploiting the reduced model and that
𝜖 ≪ 1we can more carefully analyse the stability of the SPE and the
dynamics away from steady-state.
To determine the local stability of the SPE we first reduce the model
further by writing 𝑄𝑡= 1 − 𝑃−𝑄𝑝and then compute the Jacobian, 𝐽,
asymptotically
𝐽∼𝐴+𝜖𝐵 +(𝜖2),(24)
where
𝐴=
0 −𝑃00
−𝑎
𝜆−𝑎
0𝑃00
,
𝐵=
−1 − 𝑆0
𝜆
𝑎(1+𝛤−1)0
𝑆0−
𝜆(𝜎+𝛤−1)
𝑎𝜎(1+𝛤−1 )− 1 −𝑟𝛤 −1𝑆0
0𝛤−1
𝜆
𝑎(1+𝛤−1)𝜎 𝑟𝛤 −1𝑆0− 1
,(25)
and
𝑃∗∗ ∼𝑃0=𝑟𝜎(𝛤+ 1)
𝛤+𝑟𝜎 ,(26a)
𝑆∗∗ ∼𝜖𝑆0=𝜖𝛤(1 − 𝑟𝜎)
𝑟𝜎(1 + 𝛤)(26b)
are the leading order SPE equilibrium for 𝑃∗∗ and 𝑆∗∗ given by (20)
and (23) respectively. We also define
𝜆=𝑃0−𝑎−𝑃0
𝜎= − 𝑟(𝛤+ 1)(1 − 𝜎) + 𝑎(𝛤+𝑟𝜎)
𝛤+𝑟𝜎 <0.(27)
3.2. Eigenvalues of the Jacobian matrix
We seek the eigenvalues of the Jacobian, 𝐽𝒙=𝜆𝒙given by (24)
which to leading order are the eigenvalues of 𝐴in (25). There is one
non-zero real and negative eigenvalue,
𝜆given by (27). The remain-
ing eigenvalues are zero with algebraic multiplicity 2 and geometric
multiplicity 1 with lone left and right eigenvectors,
𝒆†
1= [1,0,1]𝑇,𝒆1= [1,0,−1]𝑇(28)
respectively. The matrix 𝐴also has a generalized eigenvector
𝒆2=0,1,
𝜆
𝑎𝑇
,(29)
such that 𝐴𝒆2= −𝑃0𝒆1. Since the only non-zero eigenvalue to 𝐴is
real and negative then the stability of the SPE is determined by per-
turbative corrections to the zero eigenvalue. Since the zero eigenvalue
is degenerate then (see [50] for example) we expand
𝜆∼𝜖𝜆0+𝜖𝜆1+𝜖3∕2 𝜆2+ … ,
𝑥∼𝒆1−𝜖𝜆0
𝑃0
𝒆2+𝜖𝒙2+𝜖3∕2𝒙3+ … ,
where the expansion to (𝜖3∕2)is necessary to determine the corrections
up to (𝜖). The problem at (𝜖)is
𝐴𝒙2+𝐵𝒆1= −
𝜆2
0
𝑃0
𝒆2+𝜆1𝒆1.(30)
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Multiplying by the left-eigenvector and realizing that the left and right
eigenvectors in (28) are orthogonal leads to an expression for 𝜆0,
𝜆0= ±𝑖𝑎(1 − 𝑟𝜎)
𝜆,(31)
where we have used that
𝜆 < 0. Thus, the leading order eigenvalue is
complex and so we expect oscillatory behaviour towards steady state
with period,
𝑇∞= 2𝜋
𝜆
𝜖𝑎(1 − 𝑟𝜎).(32)
The eigenvector, 𝒙2corresponding to 𝜆0is obtained from (30) yielding,
𝒙2= − 𝜆1
𝑃0
𝒆2+𝒖,𝒖=1 − 𝑟𝜎
𝜆𝑃0
+𝑆0(1 + 𝑟𝛤 −1)
𝑎−
𝜆
𝑎𝑃 2
0
,−1
𝑃2
0
,0𝑇
.(33)
Since the leading order eigenvalue is imaginary, the dominant growth
behaviour comes from the second eigenvalue correction. At (𝜖3∕2)the
eigenvalue problem is
𝐴𝒙3−𝜆0
𝑃0
𝐵𝒆2=𝜆2𝒆1− 2 𝜆0𝜆1
𝑃0
𝒆2+𝜆0𝒖,(34)
and once again by multiplying with the left-eigenvector 𝒆†
1we furnish
an expression for 𝜆1
𝜆1= − 1
2𝑃0
+1 − 𝑟𝜎
2
𝜆2
𝜆𝑎
𝛤+ 1 −(1 − 𝜎)𝑃0
𝜎,(35)
which is real. When there is no social influence on quitting then 𝛤→∞
and the only positive term in (35) is eliminated. However,
d𝜆1
d𝑎=𝛤+
𝜆
2+ 1(1 − 𝜎)𝑃0(1 − 𝑟𝜎)
𝜎
𝜆3(𝛤+ 1)
>0,
and
𝜆∞
1= lim
𝑎→∞𝜆1= − 𝛤+𝑟2𝜎2
2𝑟𝜎(𝛤+ 1) <0,(36)
so the eigenvalue is always real and negative and the SPE is locally
asymptotically stable similar to the case 𝛤→∞as was determined by
Sharmoi and Gumel [15]. We note that,
d𝜆∞
1
d𝜎=𝛤−𝑟2𝜎2
2𝑟𝜎2(𝛤+ 1) ,(37)
which has a maximum when 𝜎=𝜎∗given by
𝜎∗=𝛤
𝑟,(38)
and so long as 𝛤 < 𝑟2then 𝜎∗<1. Therefore, when 𝛤 < 𝑟2then the
real part of the eigenvalue is least negative when 𝜎=𝜎∗. Otherwise,
it is least negative when 𝜎= 1. We note that 𝑟 < 1is required for
the Basic Reproduction Number 0>1in (7) and thus this optimum
is realized for 𝛤 ≪ 1indicating very strong social influence. We
emphasize that optimum here refers to a scenario leading to the most
persistent oscillatory behaviour.
4. Numerical simulations
When social-influence on cessation is not considered, then 𝛤→∞
and from (37) we expect that 𝜆∞
1is maximum when 𝜎= 1. Furthermore,
we can compute that
d𝜆∞
1
d𝛤= − 1 − 𝑟2𝜎2
2𝑟𝜎(𝛤+ 1)2<0,(39)
and so 𝜆∞
1becomes increasingly negative as 𝛤increases. All of this
suggests that if oscillations occur as 𝛤→∞they will appear with
large 𝜎and will dampen quickly. This is confirmed in Fig. 3, where
we simulate the reduced model (19) with parameters 𝜇= 1∕8 y−1 ,
Fig. 3. Simulation of the reduced model (19) with parameters 𝜖= 0.03125,𝑟= 0.5,
𝑎= 1000, and 𝛤→∞while varying 𝜎. We take as initial conditions [𝑃 , 𝑆, 𝑄𝑡, 𝑄𝑝] =
[1 − 𝜖, 𝜖, 0,0] and scale time with 𝛽−1 = 0.25y. Smokers (𝑆) are plotted in solid lines
while potential smokers (𝑃) are plotted in dashed line.
Table 2
Comparison of periods of oscillation of (19) as the solution approaches steady state.
The numerical period 𝑇numeric is taken as the period between the last successive local
maxima in the simulation while 𝑇∞is given by (32). The remaining parameters of
the simulation are 𝜖= 0.03125,𝑟= 0.5,𝑎= 1000, and 𝛤= 0.0625. We take as initial
conditions [𝑃 , 𝑆, 𝑄𝑡, 𝑄𝑝] = [1 − 𝜖, 𝜖, 0,0] and scale time with 𝛽−1 = 0.25y.
𝜎 𝑇 ∞(years) 𝑇numeric (years)
0.2 9.38 9.66
0.3 9.64 9.77
0.4 9.94 9.94
0.5 10.26 10.26
0.6 10.62 10.65
0.7 11.02 11.56
𝛽= 4 y−1,𝛾1= 2 y−1 , and 𝛾3= 0 y−1 yielding non-dimensional
parameters 𝜖= 0.03125 ≪1,𝑟= 0.5, and 𝛤→∞. Since the model
is insensitive to 𝛼1and from (3.2) we have d𝜆1
d𝑎>0then we take
𝑎= 1000 ≫1to exploit the large 𝑎limit.
We now introduce the social cessation influence by taking 𝛤=
0.0625. From (38) we have that 𝜎∗= 0.5and therefore we expect
oscillations to be most persistent around this value. We also suspect
from (39) oscillations to appear for smaller values of 𝜎compared to
the case of 𝛤→∞. Indeed, both of these are observed in Fig. 4
where we note the final time is more than triple that of Fig. 3. The
predicted value of the period near steady state given by (32) is in good
agreement with the numerically computed period from Fig. 4 as shown
in Table 2 where the approximate steady state period is taken as the
period between the last successive maxima prior to reaching steady
state. From (39) the oscillations should become more persistent as 𝛤
decreases (stronger social influence on cessation). This is demonstrated
in Fig. 5 where we have taken 𝛤=𝜖2= 9.766 × 10−4. From (38), this
value of 𝛤yields 𝜎∗= 0.0625. We observe that even after 500 years,
oscillations persist. For both Figs. 4 and 5oscillatory solutions emerge
early that are distinct from the terminal period. For example, in Fig. 5,
early oscillations have periods exceeding one decade which indicates
that with social influence on cessation, a social smoking contagion
can experience a long dormancy before recurrent outbreaks emerge.
Furthermore, in Fig. 5(b), we demonstrate that fairly robust oscillations
can exist for several decades. The social influence is demonstrated in
Fig. 6 where we plot smokers and permanent quitters. Peaks in smoking
occur when permanent smokers are at a local minimum.
Related but separate to oscillatory behaviour is that as social influ-
ence becomes stronger (𝛤decreases) then the time to reach steady state
increases as shown in Table 3. Therefore social influence on cessation
can significantly delay the onset to an endemic level of smoking.
Mathematical Biosciences 377 (2024) 109303
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S.I. Machado-Marques and I.R. Moyles
Fig. 4. Simulation of the reduced model (19) with parameters 𝜖= 0.03125,𝑟= 0.5,
𝑎= 1000, and 𝛤= 0.0625 while varying 𝜎. We take as initial conditions [𝑃 , 𝑆, 𝑄𝑡, 𝑄𝑝] =
[1 − 𝜖, 𝜖, 0,0] and scale time with 𝛽−1 = 0.25y. Smokers (𝑆) are plotted in solid lines
while potential smokers (𝑃) are plotted in dashed line. The scale on the smokers axis
has been reduced to visualize secondary oscillations from the initially large peak.
Table 3
Computation of the time 𝑇steady for the smoker steady state simulation of (19) to reach
𝑆0given by (26b) while varying 𝛤. The remaining parameters of the simulation are
𝜖= 0.03125,𝑟= 0.5,𝑎= 1000, and 𝜎= 0.15. We take as initial conditions [𝑃 , 𝑆, 𝑄𝑡, 𝑄𝑝] =
[1 − 𝜖, 𝜖, 0,0] and scale time with 𝛽−1 = 0.25y. The numbers should consistently decrease
with increasing 𝛤however, small oscillations may persist that are below numerical
detection.
𝛤 𝑇steady (years)
0.1 64.9
0.2 33.6
0.3 22.6
0.4 23.3
0.5 13.2
0.6 13.9
0.7 14.15
0.8 14.6
0.9 15.3
1 15.6
5. Discussion & conclusion
We have developed a deterministic model of vaping in adolescents
using literature from the social sciences to account for several types of
behavioural influences. Through analysis and simulations, we demon-
strated that the social influence from permanent quitters on causing
smokers to quit had the most influence on vaping behaviours within
the population.
We computed a sensitivity analysis on the full model (1) which
demonstrated that the parameters 𝛼1,𝛼2, and 𝛾2were not sensitive
to any of the outcomes assessed. 𝛼𝑖are the parameters associated to
relapse from temporary quitting back into smoking with 𝛼1the natural
relapse and 𝛼2the influence from smokers to persuade temporary
quitters to resume smoking. While neither were sensitive, 𝛼1had a more
structural role, weakly influencing the Basic Reproduction Number
(7). This aligns with literature suggesting that social factors are not
important in decisions to re-initiate vaping [49]. These results are also
supported by behavioural studies that show temporary quitters are not
as motivated to maintain their quitting status [45]. The parameter 𝛾2
is social influence to stop smoking from temporary quitters. Mathe-
matically, it is likely insensitive because temporary quitters are always
Fig. 5. Simulation of the reduced model (19) with parameters 𝜖= 0.03125,𝑟= 0.5,
𝑎= 1000,𝛤= 9.766 × 10−4, and 𝜎= 0.0625. We take as initial conditions [𝑃 , 𝑆 , 𝑄𝑡, 𝑄𝑝] =
[1 − 𝜖, 𝜖, 0,0] and scale time with 𝛽−1 = 0.25y. Smokers (𝑆) are plotted in blue while
potential smokers (𝑃) are plotted in red. The scale on the smokers axis has been
reduced to visualize secondary oscillations from the initially large peak. The second
plot indicates that fairly robust oscillations can persist for many years before steady
state is achieved.
transient in the model and are always available to resume smoking.
A behavioural explanation may be that smokers are less influenced by
temporary quitters as they observe their relapses and therefore do not
see it an incentive to quit smoking.
The influence of permanent quitters on cessation (𝛾3), which had
not previously been considered in models, emerged as an important
factor in the dynamic trajectory of e-cigarette usage in the adolescent
population. As a social contagion model, permanent quitters have
lifelong immunity and so that compartment directly impacts the size of
the population who can smoke. This is corroborated by the parameter
𝜎which is the proportion of quitters who become permanent. For all
response variables in Fig. 2,𝜎showed a strong negative sensitivity,
meaning that all responses decreased as more people became perma-
nent quitters. 𝛾3provides another pathway for smokers to become
permanent quitters and unlike the temporary quitters, the compartment
is static and so has more impact on model outcomes. Behaviourally,
because this compartment is static it exerts a permanent influence on
the population until they age out of the model. With this influence, it
Mathematical Biosciences 377 (2024) 109303
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S.I. Machado-Marques and I.R. Moyles
Fig. 6. Simulation of the reduced model (19) with parameters 𝜖= 0.03125,𝑟= 0.5,
𝑎= 1000,𝛤= 9.766 × 10−4, and 𝜎= 0.0625. We take as initial conditions [𝑃 , 𝑆 , 𝑄𝑡, 𝑄𝑝] =
[1 − 𝜖, 𝜖, 0,0] and scale time with 𝛽−1 = 0.25y. Smokers (𝑆) are plotted in blue while
permanent quitters (𝑄𝑝) are plotted in red. The scale on the smokers axis has been
reduced to visualize secondary oscillations from the initially large peak.
is possible that vaping becomes less normalized and curiosity about e-
cigarettes decreases, which are factors known to encourage e-cigarette
initiation [6].
We showed that the terminal outcomes of the model without social
influence on cessation presented by Sharomi and Gumel [15] did not
change, namely that the Smoking-Free Equilibrium was unstable with
the same Basic Reproduction Number, that there was a single feasible
Smoking-Present Equilibrium, and that this endemic equilibrium was
locally asymptotically stable. However, we showed that the transient
dynamics between the two models are quite different. For example,
the introduction of 𝛾3can induce significant oscillations that last for
decades and centuries which highlights an important caution when only
considering steady state considerations.
When 𝛾3= 0 (or sufficiently small) then oscillations are most robust
when 𝜎= 1. This means that the steady state to endemic smoking is
delayed the most when everyone becomes a permanent quitter. As dis-
cussed, permanent quitting confers lifelong immunity to smoking and
thus it removes an infection pathway for the social contagion. Recur-
rent smoking outbreaks occur because this group eventually ages out of
the population of social influence and is replaced by potential smokers.
We note however that while the number of peaks of smoking increase,
the peak-size decreases monotonically as 𝜎approaches 1 (see Fig. 3).
When 𝛾3increases so that 𝛤 < 𝑟2then oscillations are most robust at
a value 𝜎=𝜎∗given by (38). At large 𝜎the combination of the immu-
nity of permanent smokers as well as their strong social influence on
quitting quickly stabilizes the steady smoking population to its endemic
value. Therefore, to see more transient dynamics 𝜎needs to decrease
so that there is a balance between the strong influence of permanent
quitters and the diminishing size of their numbers. The proportion
of quitters that relapse is quite high and so typically 𝜎is close to
zero [47]. This leads to the potentially unintuitive result that strong
social influence from peers can lead to more volatile smoking dynamics
the more frequently smokers relapse. Furthermore, it means that most
real parameters likely exist within the robust oscillation regime.
Fig. 6 demonstrates that recurrent peaks in smoking are precipitated
by a loss of permanent quitters in the system. When a smoking peak
occurs, those who will permanently quit do so fairly rapidly and then
there is a slow rate of decay during smoking dormancy until the levels
get small enough that a smoking outbreak occurs. Behaviourally, the
social pressures of the permanent quitters reduce smoking rates signif-
icantly. As these permanent quitters age out of their social influence
they are replaced at a slower rate due to the low proportions of
smokers. Eventually, the overall memory of the system is lost and the
negative aspects of smoking that the permanent quitters influence are
not known to new naive potential smokers, rapidly increasing smoking
proportions. This type of behaviour has also been linked to vaccine
hesitancy, where complacency can cause people to perceive a low
amount of risk to a disease because of the historical effectiveness of
the vaccine [51].
The results of this paper focus on transient oscillations towards an
asymptotically stable steady state. We do not conjecture nor advocate
that such oscillations are present in e-cigarette usage. Certainly, tradi-
tional smoking has seen a fairly consistent decrease in usage over the
last several decades [47]. However, there have also been significant
public health and marketing campaigns against the dangers of smoking,
neither of which are considered in this model. The same report shows
more volatility in e-cigarette usage which could show that oscillations
are happening. Regardless, the actual presence of oscillatory solutions
are irrelevant to our conclusions. We demonstrated that the introduc-
tion of socially influence cessation can disrupt the dynamic structure
of a model of smoking. This provides researchers a mechanism to
understand oscillatory behaviour when it appears in a social contagion
model and also demonstrates the importance of considering transient
dynamics in model analysis and not just the stability of steady states,
which could take decades or centuries to reach.
The presence of socially-influenced cessation decreases the time to
reach the endemic steady state equilibrium as shown in Table 3. Thus,
even if oscillations are not deemed to occur in the population, this
model could be used to estimate social-influence parameters from a
historical data trend with known relapse rates. This can strengthen the
terse research on the role of behaviour in cessation.
We do not compare our model to data for several reasons. Firstly,
there is limited data on e-cigarette usage. Annual data collection on
e-cigarette prevalence in US adolescents has been occurring for ap-
proximately 10 years showing a generally increasing linear trend [52].
This data is only available for the smoking compartment. Data on
cessation is even more sparse, although some recent studies exist [53,
54]. Unfortunately, the data have few sampled time points, does not
distinguish cessation status (temporary vs. permanent quitter), and
does not explicitly link cessation attempts to reasoning (‘‘natural’’ vs.
socially-influenced). Secondly, the emphasis of our study on long-term
trends due to social influence means that vaping has not been present
long enough to confirm any of these trends. Overall, the focus of
our study is exploratory rather than explanatory, but increased and
more frequent data will potentially support future model validation.
A study from Glasgow on adolescent smoking analyzes the mechanism
of adolescent smoking to the proximity of social networks [33]. This is
an encouraging trend in data collection and analysis.
We have seen that during intermediary times far from steady state,
fairly persistent oscillations can occur. The onset of these oscillations
can behave differently than those predicted by the steady state period
(32) and therefore it is of interesting future work to analyse these
pseudo-oscillatory states. A further area of future work is to include
the influence of anti-smoking media campaigns as well as the strength
of social influence in like-minded groups. For example, smokers may
be more influenced to keep smoking by their peers than heed advice
from quitters who may exist outside their immediate social circle.
The influence from other age groups may be important to consider,
particularly in family members who may encourage smoking initiation
or even provide access to vaping products.
CRediT authorship contribution statement
Sarah I. Machado-Marques: Writing – review & editing, Writing
– original draft, Visualization, Validation, Software, Methodology, For-
mal analysis, Conceptualization. Iain R. Moyles: Writing – review &
editing, Writing – original draft, Visualization, Validation, Supervision,
Software, Project administration, Methodology, Funding acquisition,
Formal analysis, Conceptualization.
Mathematical Biosciences 377 (2024) 109303
9
S.I. Machado-Marques and I.R. Moyles
Declaration of competing interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this article.
Acknowledgements
IRM is supported by an NSERC Discovery Grant.
Appendix. Preservation of model structure
In reducing the model, it was shown that the root which vanishes
from the full model’s polynomial in 𝑆∗∗ was negative and therefore,
infeasible. It was also shown that the larger SPE of the reduced model
is always infeasible. Thus both the reduced and full models (1) and (19)
have the same feasible steady state characteristics. However, we must
also demonstrate that the entire dynamics are relatively unchanged in
the model reduction process. We plot the model comparisons in Fig. A.7
for parameters 𝜇= 1∕8 years−1,𝛽= 2 years−1,𝛼1=𝛼2= 36.5years−1,
𝛾1=𝛾2= 1 year−1,𝛾3= 1.5years−1,𝜎= 0.15. The agreement is quite
strong.
We also demonstrate that the model reduction does not affect
parameter sensitivity. The same LHS matrix from the prior sensitivity
analysis in Section 2.2 was used, but removing the parameters that
are not in the reduced model (𝛼2and 𝛾2). The PRCC results from the
sensitivity analysis are shown in Fig. A.8. Again, parameters associated
with producing smokers (𝛽, 𝛼1) were found to be positively correlated
with the response variables of 0, peak value, smokers’ equilibrium,
and final size. The correlations of 𝛼1still prove to be insignificant. The
results continue to indicate that 𝛽is the most significant factor in the
severity of e-cigarette smoking in the population. It is also the most
significant factor in when the vaping peak would occur, with a strong
negative correlation. Parameters associated with smoking cessation
(𝛾1, 𝛾3, 𝜎) were found to be negatively correlated with all response
variables, with 𝜎remaining the most significant parameter of all the
cessation associated parameters. The non-social cessation rate of 𝛾1was
once again found to be significant in relation to 5 out of the 6 response
variables. Importantly, 𝛾3was still found to have a significant effect on
the smokers’ equilibrium and final size values.
Fig. A.7. Simulation of the full model (1) (solid lines) versus reduced model (19) (dashed lines), with parameters 𝜇= 1∕8 years−1,𝛽= 2 years−1,𝛼1=𝛼2= 36.5years−1,𝛾1=𝛾2= 1
year−1,𝛾3= 1.5years−1,𝜎= 0.15 and initial conditions 𝑃(0) = 0.92,𝑆(0) = 0.08, with 0= 7.20.
Mathematical Biosciences 377 (2024) 109303
10
S.I. Machado-Marques and I.R. Moyles
Fig. A.8. PRCC analyses for the reduced model using 3000 LHS samples for the parameter ranges in Table 1, where the red horizontal lines at ±0.5indicate the threshold for
significance and * denotes a statistically significant result (𝑝-value <0.05).
Mathematical Biosciences 377 (2024) 109303
11
S.I. Machado-Marques and I.R. Moyles
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