Computational and Mathematical Methods in Medicine

Volume 2018, Article ID 3738584, 11 pages

https://doi.org/10.1155/2018/3738584

## Qualitative and Sensitivity Analysis of the Effect of Electronic Cigarettes on Smoking Cessation

Department of Mathematics, Pusan National University, Busan 46241, Republic of Korea

Correspondence should be addressed to Il Hyo Jung; rk.ca.nasup@gnujhli

Received 27 December 2017; Accepted 5 July 2018; Published 15 August 2018

Academic Editor: Ruisheng Wang

Copyright © 2018 Jae Hun Jung et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Recently, the role of the electronic cigarettes (e-cigarettes) in a way to reduce smoking is increasing. E-cigarettes are a device that delivers only the nicotine, and its use is considered less harmful to health compared with tobacco cigarettes. Smokers frequently make use of e-cigarettes as one of the nonsmoking aid devices. In this work, we propose a mathematical model to analyze the effect of e-cigarettes on smoking cessation. The stability and the bifurcation of the model have been discussed. The parameter estimations from the observed data are drawn, and using the parameters, a reasonable smoking model has been designed. Moreover, by considering the sensitivity results depending on the basic reproduction number , the effective strategies that reduce the smokers are investigated. Numerical simulations of the model show that e-cigarettes may somewhat diminish the numbers of smokers, but it does not reduce the number of quitters ultimately.

#### 1. Introduction

Smoking is well known as one of the most serious global public health problems. According to the report by the World Health Organization (WHO) [1], smoking is the legalization of a drug that kills many of its users. In other words, smoking leads to disease and disability and harms almost every organ of the body by both active and passive (second-hand) smoking. Moreover, smoking also induces an addiction, so that smokers who want to stop smoking cannot do it. Each year, active smoking is responsible for the death of about 5 million people in the world. In terms of casualties, it is more lethal than tuberculosis, human immunodeficiency virus/acquired immunodeficiency syndrome (HIV/AIDS), and malaria combined. Overall, 600,000 people are estimated to die annually due to the effects of second-hand smoke. Therefore, smoking is a serious health risk.

For that reason, the government encourages enrollment in smoking cessation programs, as well as the use of nicotine patches and nicotine gums so as to reduce the number of smokers. In addition, it establishes no smoking areas. However, since smoking is very difficult to quit, some smokers tend to use electronic cigarettes as a substitute for tobacco cigarettes. E-cigarettes are also considered to be less harmful than tobacco cigarettes [2], as, unlike cigarettes whose smoke contains thousands of harmful substances, such as tar and carbon monoxide, the e-cigarettes contain only nicotine. In fact, from the medical point of view, e-cigarettes have been studied extensively [3, 4]. For example, in [5] and [6], e-cigarettes are described as a valuable product for smoking cessation. However, in [7], the authors suggested that advertising for e-cigarettes must be banned until scientific evidence appears.

On the one hand, from the mathematical modelling point of view, the smoking cessation models have been studied by using mathematical modelling [8–11]. For instance, Sharma and Misra have studied a mathematical model of smoking cessation with media campaigns and bifurcation analysis [12]. Furthermore, Pang et al. have proposed a mathematical model with a saturated incidence rate to explore the effect of controlling smoking [13]. Similarly, Zaman has proposed several mathematical models of an type [14–16].

Although there have been many studies on smoking, to the best of our knowledge, none of previous studies have analyzed the effect of e-cigarettes on smoking cessation using a smoking model. It may be difficult to judge whether e-cigarettes actually help smoking cessation. Therefore, it is worthwhile to study a mathematical model that can identify the characteristics of nonsmokers and quitters, unlike statistical methods and experiments that confirm only the characteristics of simple smokers.

In the present study, we investigate such a smoking model based on the real data from the US Department of Health and Human Services [17]. The aim of this paper is to demonstrate both the addictive nature of smoking and the efficacy of e-cigarettes as an aid in smoking cessation using a mathematical model. Numerical simulations of the model confirm the dynamics of two aspects of the effect of e-cigarettes: the numbers of smokers and nonsmokers.

The rest of the article is composed as follows: In Section 2, a mathematical model to assess the effect of e-cigarettes on smoking cessation is proposed; furthermore, the dynamics of the model, the basic reproduction number, and the stability of equilibria are investigated. In particular, the condition for occurring bifurcation situation is presented. In Section 3, the parameter estimations from real data are provided. The sensitivity analysis and numerical simulation confirm the results that are obtained analytically. Finally, the conclusions are briefly summarized in Section 4.

#### 2. Mathematical Model and Analysis

##### 2.1. Model

The traditional epidemiologic model of agent, host, vector, and environment is useful for studying the interplay of various influences on patterns of tobacco use in populations. Since the dynamics of smoking cause disease and addiction that is defined when occurring craving, tolerance, withdrawal symptoms, and loss of control, it is very similar to that of an epidemic [18]. For example, a nonsmoker comes into contact with smokers and starts smoking from his influence. Based on epidemic models, therefore, we propose a nonlinear mathematical model to assess the effect of e-cigarettes on smoking cessation.

First, we set up a region with the total population *N* at time *t*. The total population is divided into four subpopulations [19]: potential smokers (*P*) who do not smoke yet but might become smokers in the future; smokers (*S*) who have smoked more than 100 cigarettes in their lifetime and reported smoking “everyday” or “some days” at the time; e-smokers (*E*) who currently use e-cigarettes; and quitters (*Q*) who had smoked more than 100 cigarettes in their lifetime and reported smoking “not at all” at the time. We assume that all of the natural death rates equal to the birth rates is *μ*, and the mortality rate due to the specific diseases that caused by smoking is not considered.

We have considered the effective contact rate, that is, the average number of visits to social gatherings of potential smokers for influential contact of smokers per unit time and the probability of the proportion of becoming smokers after a casual smoking. The constant is the effective contact rate that potential smokers can become smokers due to peer influence and come in the smokers class at rate . The constant is the effective contact rate, but this case is a relapse to smoking by peer influence. Then quitters can become smokers due to peer influence at rate . In this part, we have considered the effects of peer influence in dynamics of smoking and presented the nonlinear term. Furthermore, we assume that the start smoking rate is greater than the relapse rate because the influence of the curiosity of the people who has not smoked is more than people who quit smoking.

To quit smoking, however, we consider two kinds of groups: smokers who may quit smoking by their own will at rate or smokers who use e-cigarettes at rate . In the latter case, e-smokers who used e-cigarettes are likely to revert to smokers at rate *α*. The constant represents the per capita quit rate that stops smoking by use of e-cigarettes. The model parameters are summarized in Table 1. The proposed model is shown in Figure 1 and is described as follows:where the initial conditions are given by