Mathematical analysis of giving up smoking model via harmonic mean type incidence rate

doi:10.1016/j.amc.2019.01.053

Applied Mathematics and Computation

Volume 354, 1 August 2019, Pages 128-148

https://doi.org/10.1016/j.amc.2019.01.053 Get rights and content

Abstract

This article deals with qualitative analysis of a smoking model and provides parametric conditions for controlling diseases under the influence of smoking. The model is obtained by taking into account a novel uptake function, which relates the incidence of potential smoker with occasional smoker, of harmonic mean type for the potential and occasional smokers. Introducing a new type of incidence rate, the local as well as global stabilities for the proposed model are discussed at its steady–states. Minimizing the number of individuals involved in a substance abuse in any community is very difficult task, therefore we have introduced four control variables in the controlled system. Pontrygin’s maximum principal is used to derive optimality system. Finally, the obtained results are illustrated numerically and graphically.

Introduction

Among the natural disasters epidemic diseases kill many people in different eras. Some of these diseases are now curable/controllable and scientists have made enormous efforts to control such diseases. For example, for the polio control free vaccination is provided to kids of five years. These epidemics are natural and could be treated up to some extent. Dengue, malaria, small pox etc are the epidemics for which the pharmacists have manufactured some medicines. Furthermore, scientists are struggling continuously to formulate more economical and effective antidotes.

In the thick of these widespread diseases, tobacco epidemic is one of the huge health hazards the world has ever faced. This is among those threats which are generated by humans themselves. Moreover, it is under the control of humans and hence by suitable strategies and efforts it is possible to curb this intimidation.

Smoking kills up to 50 percent of its users. Nearly, each year 6 million people die from smoking, of whom more than 5 million are users and ex-users and more than 0.6 million are nonsmokers exposed to passive smoking. People who die prematurely deprive their families of income, consumes family budget, raise the cost of health care and retard economic development. Death statistics reveal, there is a death caused by tobacco every 8 second; ten percent of the adult population die because of smoking related diseases. Due to the increase in smokers community, tobacco use is also a fatal threat which need to be curbed.

Mathematical modeling is a way to study the dynamics of many real world problems. It is the proficiency of converting problems from an application area into tractable mathematical formulations whose theoretical and numerical analysis provide insight and guidance which are useful for the originating application. A mathematical model is an abstract model that uses mathematical language to describe the behavior of a system. In mathematical modeling recently many people have contributed significantly; therefore, we refer the readers to see the manuscripts and references there in [1], [2], [3]. Various infectious diseases are modeled mathematically. Similarly the dynamics of smoking has modeled mathematically. Furthermore, in order to perceive the spread of smoking and to predict the impact of smoking on society and to reduce the number of smokers, it is worth noticing to study the dynamical aspects of smoking in a community.

In order to identify the spreading rate of smoking using mathematical tools, researchers have introduced different classes of smokers. In 1997, Castillo-Garrsow et al. [4], presented a general epidemiological model to describe dynamical aspects of giving up smoking model. The authors assumed PSQ model with a total constant population divided into three categories: potential smoker P, smokers S and quit smokers Q. Moreover, they considered the effect of peer pressure, relapse, counseling and treatment. Subsequently, in 2008 Sharomi and Gumel [5], introduced a new category Q_t of smokers who temporarily give up smoking. For more detail on giving up smoking model we refer to see [6], [7], [8], [9], [11].

In epidemiology, incidence is the probability of occurrence of a specific medical condition in a population within a specified period of time. In the transmission of smoking epidemics incidence rate plays a vital role. In order to explain the dynamical aspects of various individual in the community, researchers have incorporated bi-linear and saturated incidence rates [4], [12], [13]. In a finite time these aforementioned incidence rates do not help to show the credibility of having the smokers population tends to extinction. For this purpose the authors in [11] incorporated square root type incidence rate, which was introduced by Mickens [13].

In order to approach the smokers population to extinction rapidly, we introduce harmonic mean type of incidence rate between occasional and potential smokers. The relation between the two incidence rates is provided in the main part of the manuscript. The paper is organized as follows. Section 2 contains some basic biological aspects of the proposed model. Sections 2.1 and 2.2 provide results related to basic reproductive number and local stability of the model at two different equilibrium points. Section 3 gives global dynamics for the system (2.7). Numerical results are provided in Section 2.2 to elaborate the main results. Section 4.1 is reserved for the study of optimal control. Concluding remarks are given at the end of the paper to show the novelty of the present work.

Section snippets

Formulation of the model and main results

Modeling the dynamics of giving up smoking model, we focus on the population. Assume that the total population size at any time t is represented by N(t). The population N(t) is partitioned into four classes P(t) (potential smokers), L(t) (occasional smokers), S(t) (chain smokers) and Q(t) (quit smokers). The governing equations for the proposed model are given as [11]: $\begin{matrix} \frac{d P}{d t} & = λ - β W (L, P) - (d + μ) P \end{matrix}$ $\begin{matrix} \frac{d L}{d t} & = β W (L, P) - (ζ + d + μ) L \end{matrix}$ $\begin{matrix} \frac{d S}{d t} & = ζ L - (δ + d + μ) S \end{matrix}$ $\begin{matrix} \frac{d Q}{d t} & = δ S - (μ + d) Q . \end{matrix}$

Furthermore, W(L, P) is a functions know as uptake

Global stability analysis

This section is devoted to study the global stability analysis of model (2.7) for smoking free as well as smoking present equilibrium points. The method of Castillo-Chavez [19] is used to prove the global asymptotically stable at smoking free equilibrium. While to show that the model (2.7) is globally asymptotic stability at smoking present equilibrium, the geometrical approach is used as main tool [20].

First, we provide a brief analysis of the Castillo-Chavez method and geometrical approach to

Optimal control strategy

The usefulness of a mathematical model is assured if the system can be acted on by the use of “control” parameters. Optimization or optimal control of biological phenomena becomes possible, when applied to drugs, for example the possibility of optimal therapeutics. This step consisting of associating a model with a biological system which is difficult, due to lack of physical knowledge about the phenomenon. Indeed, most of the time we have only experimental data and some results from literature

The improved GSS1 method

Before the formulation of GSS1 method for our model, it is instructive to start with the conservation law (2.3) of the total population N for the scheme is given by Gumel et al. [32]. $\begin{matrix} \frac{N_{n + 1} - N_{n}}{h} \leq λ - μ N_{n} . \end{matrix}$

Given (P_n, L_n, S_n, Q_n) ∈ D, let $N_{n} = P_{n} + L_{n} + S_{n} + Q_{n}$ . The GSS1 scheme for the system (4.1) is given in the sequel.

The resolution of the optimality system is created improving the Gauss-–Seidel-like implicit finite-difference method developed by Gumel et al. [32] denoted as GSS1. This method initially

Discussion on numerical simulations

In this section, our theoretical discussions are illustrated through numerical simulations by taking into account some realistic data based on the mathematical and survey study conducted by Pang et al. [33] in China. As a tobacco victimized country in the world, China is being burdened by smoking-related illnesses. China confronts a daunting challenge in controlling smoking. The country grows more tobacco, produces more cigarettes, makes more profits, and has more smokers than any other country

Concluding remarks

In the present paper dynamical aspects of giving up smoking model are studied. We have introduced the contact of potential smokers P and occasional smokers L in a new format, which is designated as harmonic mean type incidence rate. In the beginning, people start smoking occasionally for a variety of different reasons. Some think it looks cool. Others start because their family members or friends smoke. Statistics show that about 9 out of 10 tobacco users start occasionally and then gradually

Competing interests

All authors declare that they have no competing interests.

Authors contributions

All authors carried out the proofs. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.

Acknowledgments

A sincere thanks to Dr. Sheraz Ali (Lecturer Department Economics and Development studies) and Mr. Iftikhar Ahmad(Lecturer Department of Mathematics and Statistics) University of Swat, for their diligent proofreading and English corrections of this manuscript.

References (37)

J. Singh et al.
A fractional epidemiological model for computer viruses pertaining to a new fractional derivative
Appl. Math. Comput.
(2018)
D. Kumar et al.
A modified numerical scheme and convergence analysis for fractional model of Lienards equation
J. Comput. Appl. Math.
(2018)
O. Sharomi et al.
Curtailing smoking dynamics: a mathematical modeling approach
Appl. Math. Comput.
(2008)
A. Ahlbom
Bio-statistics for Epidemiologists
(1993)
LiM.Y. et al.
A geometric approach to global-stability problems
SIAM J. Math. Anal.
(2006)
L. Bauld, The Impact of smoke-free legislation in England: Evidence review, Department of Health 2011 Bath...
A.B. Gumel et al.
A mathematical model for the dynamics of HIV-1 during the typical course of infection
Nonlinear Anal.
(2001)
The wall street journal, try try again, china announces indoor smoking ban, 2011,...
G. Zaman
Qualitative behavior of giving up smoking models
Bull. Malays. Math. Sci. Soc.
(2011)
J. Singh et al.
On the analysis of chemical kinetics system pertaining to a fractional derivative with Mittag–Leffler type kernel
Chaos
(2017)

C. Castillo-Garsow et al.

Mathematical models for the dynamics of tobacco use, recovery and relapse

Technical Report Series, BU-1505, Ithaca NY, USA

(1997)

J. Singh et al.

A new fractional model for giving up smoking dynamics

Adv. Differ. Equ.

(2017)

Z. Alkhundhri et al.

Global dynamics of a mathematical model on smoking

ISRN Appl. Math. 2014

(2014)

W.H. Fleming et al.

Deterministic and Stochastic Optimal Control

(1975)

G.u. Rahman et al.

Threshold dynamics and optimal control of an age-structured giving up smoking model

Nonlinear Anal. RWA.

(2018)

HuoH.F. et al.

Influence of relapse in a giving up smoking model

Abstr. Appl. Anal.

(2013)

DinQ. et al.

Qualitative behavior of a smoking model

Adv. Differ. Equ.

(2016)

V. Capasso et al.

A generalization of the Kermack–Mckendrick deterministic epidemic model

Math. Biosci.

(1978)

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Mathematical analysis of giving up smoking model via harmonic mean type incidence rate

Abstract

Introduction

Section snippets

Formulation of the model and main results

Global stability analysis

Optimal control strategy

The improved GSS1 method

Discussion on numerical simulations

Concluding remarks

Competing interests

Authors contributions

Acknowledgments

Appl. Math. Comput.

J. Comput. Appl. Math.

Appl. Math. Comput.

SIAM J. Math. Anal.

Nonlinear Anal.

Bull. Malays. Math. Sci. Soc.

On the analysis of chemical kinetics system pertaining to a fractional derivative with Mittag–Leffler type kernel

Chaos

Mathematical models for the dynamics of tobacco use, recovery and relapse

Technical Report Series, BU-1505, Ithaca NY, USA

A new fractional model for giving up smoking dynamics

Adv. Differ. Equ.

Global dynamics of a mathematical model on smoking

ISRN Appl. Math. 2014

Deterministic and Stochastic Optimal Control

Threshold dynamics and optimal control of an age-structured giving up smoking model

Nonlinear Anal. RWA.

Influence of relapse in a giving up smoking model

Abstr. Appl. Anal.

Qualitative behavior of a smoking model

Adv. Differ. Equ.

A generalization of the Kermack–Mckendrick deterministic epidemic model

Math. Biosci.