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The authors have declared that no competing interests exist.

Analyzed the data: JG RPPPG HLJvdM. Wrote the paper: JG.

The intake of nicotine by smoking cigarettes is modelled by a dynamical system of differential equations. The variables are the internal level of nicotine and the level of craving. The model is based on the dynamics of neural receptors and the way they enhance craving. Lighting of a cigarette is parametrised by a time-dependent Poisson process. The nicotine intake rate is assumed to be proportional with the parameter of this stochastic process. The effect of craving is damped by a control mechanism in which awareness of the risks of smoking and societal measures play a role. Fluctuations in this damping may cause transitions from smoking to non-smoking and vice versa. With the use of Monte Carlo simulation the effect of abrupt and gradual cessation therapies are evaluated. Combination of the two in a mixed scheme yields a therapy with a duration that can be set at wish.

Nicotine addiction is a significant worldwide health problem. We present a dynamical model that focuses on three key variables that play a role in the development and persistence of smoking addiction. The purpose is not to model each and every (neurobiological and psychological) detail of the processes that underlie the addiction, but to abstract away from low-level dynamics to the dynamics of these three summary variables that capture some of the most important aspects of smoking addiction. The advantage of doing so lies in that we provide a level of explanation of prominent phenomena observed in nicotine cessation research. This approach promotes the understanding of transitions in smoking behaviour in terms of some well understood mechanisms from mathematical bifurcation theory applied to the attractor dynamics of a nonlinear system.

Nicotine addiction arises from the dynamics of specific receptors on the membrane of neurons in the brain

Based on the above description of the driving mechanism, that makes people continue to smoke, we formulate the elements that will constitute a dynamical model of the state of an individual with respect to the smoking habit. We identify the state of a person by two variables:

Craving is a mental state resulting from neural processes that can be seen as dopamine-gated learning

Assuming a fixed urge to light a cigarette (solid) with λ = 0.68 (

The intake rate of nicotine is a function of

The equilibria are at the intersection of the two nullclines,

If

(a) The intake

The vertical axis give the fraction of successes out of 100 realisations. Success means that during one year after ending the therapy the person has a nicotine intake that stays below 5% of the intake before the therapy.

The variables ^{+} denotes that

The values of the rate coefficients ^{−3}[1/hr]. Eq.(2a) describes the change of the nicotine amount in the body and Eq.(2b) that of the craving intensity. The values of the parameters ^{−3}[1/(mgxh)]. Thus, we have

^{−3} and ^{−3}.

It is followed by the interval [_{s}_{s}_{s}

Since the rate coefficients

The essential element in our modelling of transitions of smoking behaviour is the presence of two or three equilibria for (2ab) which only may occur in nonlinear systems. Two of the equilibria may get unstable when the parameter

If _{0} and a diffusion term generating a random change in ^{2} ^{−3}. The distribution of realizations of

Because of this form of the stationary distribution we replace the parameter

There is a huge literature on cessation strategies. Gradual cessation and abrupt cessation strategies are compared in _{0} for which we may expect a successful treatment. From _{0} = 0.05, because then the system is bi-stable. A person being a steady smoker may be transformed into a steady non-smoker. At an earlier stage he became a smoker due to low _{0}-values (

We assume that at the start of the therapy (_{0} = 0.05 with an average nicotine intake of 0.262[mg/h]. For _{0} = 0.1 and there is no nicotine intake at all (_{0} and

The two types of treatment each have their (dis)advantages: the difference in the required length of the therapy is quite large: the gradual cessation needs much more time. Thus, abrupt cessation seems to be the best strategy _{s}_{s}_{s}_{s}_{s}_{s}_{s}

From a set of basic assumptions on nicotine intake, (self)control and craving we derived a dynamical model for smoking consisting of a set of two coupled nonlinear differential equations. For the chosen values of the model parameters three dynamical regimes may occur depending on the control parameter _{s}

In our modelling of the interaction between nicotine intake and craving we made a number of assumptions. Nonlinear functional relations are essential in grasping the essentials of the process. In our model the first term in the right hand side of both equations (2a) and (2b) are responsible for the possibility of having a bi-stable system. Under a fluctuating control the phenomenon of repeatedly resuming smoking shows up in a realistic way in the simulation (

The present model only applies to the intake of nicotine and does not help to qualify existing therapies on other elements of the treatment, such as offering substituting goodies or increasing the awareness of the risks of smoking. These, of course, do play a role in a successful completion of a treatment and in the risk of a restart. Administration of nicotine in other ways than by regular cigarettes needs in our model to be included in the total nicotine intake and is only allowed in the phase of gradual decrease.

The model may be extended in several directions, in order to account for other empirical phenomena such as, social influences on cessation success

(DOCX)

_{4}β

_{2}nicotinic acetylcholine receptors