A new fractional model for giving up smoking dynamics
- Jagdev Singh^{1}Email author,
- Devendra Kumar^{1},
- Maysaa Al Qurashi^{2} and
- Dumitru Baleanu^{3, 4}
https://doi.org/10.1186/s13662-017-1139-9
© The Author(s) 2017
Received: 8 December 2016
Accepted: 9 March 2017
Published: 23 March 2017
Abstract
The key purpose of the present work is to examine a fractional giving up smoking model pertaining to a new fractional derivative with non-singular kernel. The numerical simulations are conducted with the aid of an iterative technique. The existence of the solution is discussed by employing the fixed point postulate, and the uniqueness of the solution is also proved. The effect of various parameters is shown graphically. The numerical results for the smoking model associated with the new fractional derivative are compared with numerical results for a smoking model pertaining to the standard derivative and Caputo fractional derivative.
Keywords
MSC
1 Introduction
These days, smoking is one of the major health problems in the world. More than 5 million deaths in the world are caused due to the effect of smoking in different organs of human body. A chance of heart attack is 70% more in smokers compared to the persons who are not smoking. Smokers have a 10% higher incidence rate of lung cancer than that of non-smokers. Bad breath, stained teeth, high blood pressure, coughing are the main effects of short-term smoking. In recent years, mouth cancer, throat cancer, lung cancer, gum disease, heart disease, stomach ulcers are the main threatening due to long-term smoking. The life of smokers is 10 to 13 years shorter than that of non-smokers. Smoking kills many individuals in their most active life according to the reports of WHO. Every scientist, doctor and mathematician tries to control smoking for securing the life expectancy of an individual. To give the best representation of cigarette smoking phenomena, mathematicians tried to make different effective smoking models. The different smoking models were proposed by several authors; for example Erturk et al. [1] investigated a giving up smoking model associated with the Caputo fractional derivative, Zaman [2] analyzed the optimal campaign in the smoking dynamical system, Zaman [3] studied the qualitative response of dynamics of giving up smoking, Lubin and Caporaso [4] discussed cigarette smoking and lung cancer, Garsow et al. [5] examined the mathematical description of the dynamics of tobacco use, recovery and relapse, Sharomi and Gumel [6] demonstrated the curtailing smoking dynamics, Zeb et al. [7] investigated a fractional giving up smoking model, Alkhudhari et al. [8] analyzed the global dynamics of mathematical equations describing smoking, Khalid et al. [9] explained the fractional mathematical model of giving up smoking and many others.
Fractional calculus is applied in different directions of physics, mathematical biology, fluid mechanics, electrochemistry, signal processing, viscoelasticity, finance and in many more. In the branch of fractional calculus, fractional derivatives and fractional integrals are important aspects. Recently, many researchers and scientists have analyzed issues in this special branch [10–22]. Caputo [10] introduced a fractional derivative which allows the conventional initial and boundary conditions associated with the real world problem. Baleanu et al. [11] reported new advances in nanotechnology and fractional calculus and related issues in their monograph. Kilbas et al. [12] presented basic concepts of fractional differential equations and their applications. Bulut et al. [13] studied the differential equations of arbitrary order by making use of analytical techniques. Atangana and Alkahtani [14] examined the Keller-Segel model pertaining to a fractional derivative having non-singular kernel. Atangana and Alkahtani [15] studied a fractional non-homogeneous heat model. Singh et al. [16] explained a fractional biological population model. Kumar et al. [17] analyzed the local fractional Klein-Gordon equations. Singh et al. [18] investigated fractional coupled Burgers equations. Kumar et al. [19] presented the numerical solution of a differential-difference equation of arbitrary order having applications in nanotechnology. Area et al. [20] studied the fractional order ebola epidemic model. Carvalho and Pinto [21] analyzed a delay fractional order model for the co-infection of malaria and HIV/AIDS. Huang et al. [22] studied a novel use of the fractional logistic map. Ma et al. [23] reported new results for multidimensional diffusion equations pertaining to local fractional derivative. Kumar et al. [24] analyzed a logistic equation involving a new fractional derivative having a non-singular kernel. Kumar et al. [25] studied a modified Kawahara equation pertaining to a fractional derivative with non-singular kernel.
In a very recent attempt, Caputo and Fabrizio [26] propounded a novel fractional derivative having exponential kernel and in addition Losada and Nieto [27] analyzed the properties of a newly presented fractional derivative. The classical fractional derivatives, especially the Caputo and Riemann derivatives, have their own limitation because their kernel is singular. Since the kernel is employed to describe the memory effect of the physical system, it is obvious that due to this weakness, both derivatives cannot precisely describe the full effect of the memory. Therefore, we use the novel Caputo-Fabrizio (CF) fractional derivative to study the giving up smoking model and explain this problem in a better and more efficient manner.
The key objective of this work is to use the new fractional derivative in the giving up smoking model and imparting the details of the exactness and uniqueness of the solution by applying the fixed point theorem. The development of this article is as follows. In Section 2, the CF fractional order derivative is discussed. In Section 3, the fractional smoking model and approximate solution pertaining to novel CF fractional derivative is discussed. In Section 4, the existence and uniqueness of system of solutions is proved with the aid of the fixed point theorem. Results and discussion are given in Section 5. Lastly in Section 6, the concluding remarks are presented.
2 Preliminaries
In the present part, we give some definitions and properties of the fractional derivative as suggested by Caputo and Fabrizio [26].
Definition 2.1
Remark 1
Definition 2.2
3 Model description and giving up smoking model with a fractional derivative involving non-singular kernel
In the above system (8) b indicates the contact rate between smokers who smoke occasionally and potential smokers, a denotes the rate of natural death, c stands for the contact rate between smokers who smoke occasionally and temporary quitters, f represents the contact rate between temporary quitters who return back to smoking and smokers, d indicates the rate of giving up smoking, \((1 - e)\) stands for the fraction of smokers who temporarily give up smoking (at a rate d), e denotes the remaining fraction of smokers who give up smoking forever (at a rate d).
4 Existence and uniqueness of a system of solutions of smoking model
Theorem 4.1
Proof
On consideration of the above results, we may present the subsequent theorem.
Theorem 4.2
Proof
Now, we prove the uniqueness of a system of solutions of equation (9).
Theorem 4.3
Proof
Therefore, we verified the uniqueness of the system of solutions of equations (9). □
5 Numerical results and discussions
Comparison between standard derivative, Caputo fractional derivatives and Caputo-Fabrizio fractional derivative for the potential smokers \(\pmb{P(t)}\)
t | Standard derivative ( ρ = 1) | Caputo derivative ( ρ = 0.95) | Caputo-Fabrizio derivative ( ρ = 0.95) |
---|---|---|---|
1 | 0.6124398619 | 0.6126334075 | 0.6124401378 |
2 | 0.6218771047 | 0.6216116071 | 0.6214032210 |
3 | 0.6312904730 | 0.6303312432 | 0.6303433323 |
4 | 0.6406487116 | 0.6388461359 | 0.6392336738 |
5 | 0.6499205647 | 0.6471669258 | 0.6480474489 |
Comparison between standard derivative, Caputo fractional derivatives and Caputo-Fabrizio fractional derivative for the occasional smokers \(\pmb{L(t)}\)
t | Standard derivative ( ρ = 1) | Caputo derivative ( ρ = 0.95) | Caputo-Fabrizio derivative ( ρ = 0.95) |
---|---|---|---|
1 | 0.2416136687 | 0.2416475729 | 0.2416146612 |
2 | 0.2432477561 | 0.2432024968 | 0.2431675083 |
3 | 0.2449020824 | 0.2447359790 | 0.2447386087 |
4 | 0.2465764697 | 0.2462618874 | 0.2463278099 |
5 | 0.2482707411 | 0.2477865184 | 0.2479349601 |
Comparison between standard derivative, Caputo fractional derivatives and Caputo-Fabrizio fractional derivative for the heavy smokers \(\pmb{S(t)}\)
t | Standard derivative ( ρ = 1) | Caputo derivative ( ρ = 0.95) | Caputo-Fabrizio derivative ( ρ = 0.95) |
---|---|---|---|
1 | 0.1026239640 | 0.1025641835 | 0.1026437765 |
2 | 0.09937450414 | 0.09948799415 | 0.09956153608 |
3 | 0.09646354464 | 0.09680263691 | 0.09678199000 |
4 | 0.09382300970 | 0.09440789173 | 0.09424677174 |
5 | 0.09138482355 | 0.09222574223 | 0.09189751483 |
Comparison between standard derivative, Caputo fractional derivatives and Caputo-Fabrizio fractional derivative for the temporary quitters \(\pmb{Q(t)}\)
t | Standard derivative ( ρ = 1) | Caputo derivative ( ρ = 0.95) | Caputo-Fabrizio derivative ( ρ = 0.95) |
---|---|---|---|
1 | 0.03376048059 | 0.03377405752 | 0.03374711339 |
2 | 0.03464368108 | 0.03460575042 | 0.03458444010 |
3 | 0.03531175962 | 0.03521909598 | 0.03523042414 |
4 | 0.03582687433 | 0.03568394356 | 0.03573835831 |
5 | 0.03625118333 | 0.03606106299 | 0.03616153544 |
Comparison between standard derivative, Caputo fractional derivatives and Caputo-Fabrizio fractional derivative for the permanently quitters \(\pmb{R(t)}\)
t | Standard derivative ( ρ = 1) | Caputo derivative ( ρ = 0.95) | Caputo-Fabrizio derivative ( ρ = 0.95) |
---|---|---|---|
1 | 0.02113719363 | 0.02107624005 | 0.02112963057 |
2 | 0.02400835647 | 0.02369345384 | 0.02385493816 |
3 | 0.02674146144 | 0.02611744835 | 0.02645645955 |
4 | 0.02935448146 | 0.02839320512 | 0.02894960428 |
5 | 0.03186538945 | 0.03055013348 | 0.03134978188 |
6 Concluding remarks
In this paper, the smoking model is analyzed with the Caputo-Fabrizio derivative and makes use of the utilities of fractional calculus. To demonstrate the existence and uniqueness of a system of solutions the fixed point theorem is discussed. By employing an iterative perturbation method the special solution is obtained for the fractional order model. To present the effect of fractional order some numerical simulations are performed. By simulation it is clear that when ρ tends to 1 the CF derivative shows a more interesting behavior. Hence, it can be concluded that the newly fractional derivative is very important for modeling real world problems.
Declarations
Acknowledgements
The authors extend their appreciation to the International Scientific Partnership Program ISPP at King Saud University for funding this research work through ISPP# 63.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Erturk, VS, Zaman, G, Momani, S: A numeric analytic method for approximating a giving up smoking model containing fractional derivatives. Comput. Math. Appl. 64, 3068-3074 (2012) MathSciNetMATHView ArticleGoogle Scholar
- Zaman, G: Optimal campaign in the smoking dynamics. Comput. Math. Methods Med. 2011, Article ID 163834 (2011). MathSciNetMATHView ArticleGoogle Scholar
- Zaman, G: Qualitative behavior of giving up smoking models. Bull. Malays. Math. Soc. 34, 403-415 (2011) MathSciNetMATHGoogle Scholar
- Lubin, JL, Caporaso, ZE: Cigarette smoking and lung cancer: modeling total exposure and intensity. Cancer Epidemiol. Biomark. Prev. 15, 517-523 (2006) View ArticleGoogle Scholar
- Garsow, CC, Salivia, GJ, Herrera, AR: Mathematical Models for the Dynamics of Tobacoo use, recovery and relapse. Technical Report Series BU-1505-M, Cornell University, UK (2000) Google Scholar
- Sharomi, O, Gumel, AB: Curtailing smoking dynamics: a mathematical modeling approach. Appl. Math. Comput. 195, 475-499 (2008) MathSciNetMATHGoogle Scholar
- Zeb, A, Chohan, I, Zaman, G: The homotopy analysis method for approximating of giving up smoking model in fractional order. Appl. Math. 3, 914-919 (2012) View ArticleGoogle Scholar
- Alkhudhari, Z, Al-Sheikh, S, Al-Tuwairqi, S: Global dynamics of a mathematical model on smoking. Appl. Math. 2014, Article ID 847075 (2014). MathSciNetMATHGoogle Scholar
- Khalid, M, Khan, FS, Iqbal, A: Perturbation-iteration algorithm to solve fractional giving up smoking mathematical model. Int. J. Comput. Appl. 142, 1-6 (2016) View ArticleGoogle Scholar
- Caputo, M: Elasticita e Dissipazione. Zanichelli, Bologna (1969) Google Scholar
- Baleanu, D, Guvenc, ZB, Machado, JAT: New Trends in Nanotechnology and Fractional Calculus Applications. Springer, Dordrecht (2010) MATHView ArticleGoogle Scholar
- Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) MATHGoogle Scholar
- Bulut, H, Baskonus, HM, Belgacem, FBM: The analytical solutions of some fractional ordinary differential equations by Sumudu transform method. Abstr. Appl. Anal. 2013, Article ID 203875 (2013). MathSciNetMATHGoogle Scholar
- Atangana, A, Alkahtani, BT: Analysis of the Keller-Segel model with a fractional derivative without singular kernel. Entropy 17, 4439-4453 (2015) MathSciNetMATHView ArticleGoogle Scholar
- Atangana, A, Alkahtani, BT: Analysis of non-homogenous heat model with new trend of derivative with fractional order. Chaos Solitons Fractals 89, 566-571 (2016) MathSciNetView ArticleGoogle Scholar
- Singh, J, Kumar, D, Kilichman, A: Numerical solutions of nonlinear fractional partial differential equations arising in spatial diffusion of biological populations. Abstr. Appl. Anal. 2014, Article ID 535793 (2014). MathSciNetGoogle Scholar
- Kumar, D, Singh, J, Baleanu, D: A hybrid computational approach for Klein-Gordon equations on Cantor sets. Nonlinear Dyn. (2016). doi:10.1007/s11071-016-3057-x Google Scholar
- Singh, J, Kumar, D, Swroop, R: Numerical solution of time- and space-fractional coupled Burgers equations via homotopy algorithm. Alex. Eng. J. 55, 1753-1763 (2016) View ArticleGoogle Scholar
- Kumar, D, Singh, J, Baleanu, D: Numerical computation of a fractional model of differential-difference equation. J. Comput. Nonlinear Dyn. 11, 061004 (2016) View ArticleGoogle Scholar
- Area, I, Batarfi, H, Losada, J, Nieto, JJ, Shammakh, W, Torres, A: On a fractional order Ebola epidemic model. Adv. Differ. Equ. (2015). doi:10.1186/s13662-015-0613-5 MathSciNetMATHGoogle Scholar
- Carvalho, A, Pinto, CMA: A delay fractional order model for the co-infection of malaria and HIV/AIDS. Int. J. Dyn. Control (2016). doi:10.1007/s40435-016-0224-3 Google Scholar
- Huang, LL, Baleanu, D, Wu, GC, Zeng, SD: A new application of the fractional logistic map. Rom. J. Phys. 61, 1172-1179 (2016) Google Scholar
- Ma, M, Baleanu, D, Gasimov, YS, Yang, XJ: New results for multidimensional diffusion equations in fractal dimensional space. Rom. J. Phys. 61, 784-794 (2016) Google Scholar
- Kumar, D, Singh, J, Qurashi, MA, Baleanu, D: Analysis of logistic equation pertaining to a new fractional derivative with non-singular kernel. Adv. Mech. Eng. 9(2), 1-8 (2017) View ArticleGoogle Scholar
- Kumar, D, Singh, J, Baleanu, D: Modified Kawahara equation within a fractional derivative with non-singular kernel. Therm. Sci. (2017). doi:10.2298/TSCI160826008K Google Scholar
- Caputo, M, Fabrizio, M: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1, 73-85 (2015) Google Scholar
- Losada, J, Nieto, JJ: Properties of the new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1, 87-92 (2015) Google Scholar
- Boyd, JP: Padè approximants algorithm for solving nonlinear ordinary differential equation boundary value problems on an unbounded domain. Comput. Phys. 11, 299-303 (1997) View ArticleGoogle Scholar