Non-standard finite difference scheme and analysis of smoking model with reversion class

Zeb, Anwar; Alzahrani, Abdullah

doi:10.1016/j.rinp.2020.103785

“…Different scholars have studied the mathematical model of smoking with integer-order derivatives [9][10][11][12][13][14][15][16][17][18] and noninteger-order derivatives [19][20][21][22][23]. In this article, we modify and extend the work in [9] by adding one compartment (people exposed to thirdhand smoke) and considering the new model with non-integer-order derivatives, specifically the Caputo-Fabrizio derivative.…”

Section: Introductionmentioning

confidence: 99%

A Fractional Model for the Dynamics of Smoking Tobacco Using Caputo–Fabrizio Derivative

Melkamu

¹

,

Mebrate

²

2022

Journal of Applied Mathematics

2

0

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In this paper, we propose a Caputo–Fabrizio fractional derivative mathematical model consisting of smoker, people exposed to secondhand smoker, people exposed to thirdhand smoker, and quitters. Secondhand smoke exposure consists of an unintentional inhalation of smoke that occurs close to people smoking and/or in indoor environments where tobacco was recently used, and thirdhand smoke consists of pollutants that remain on surfaces and in dust after tobacco has been smoked, are reemitted into the gas phase, or react with other compounds in the environment to form secondary pollutants. The solution of the proposed model, which is carried out using a fixed-point theorem and an iterative method, exists and is unique. Furthermore, the model is biologically meaningful, that is, positive and bounded. The reproduction number R 0 is determined from the model. If R 0 < 1 , the smoking-free equilibrium point is asymptotically stable, and if R 0 > 1 , the smoking-free equilibrium point is unstable. The results confirm that the smoking-free equilibrium point becomes increasingly stable as the fractional order is increased. Numerical simulations are performed using a three-step Adams-Moulton predictor-corrector method for a range of fractional orders to show the effects of varying the fractional order and to support the theoretical results.

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“…In this work, we present an existing model for smoking with relapse habits with randomness and irregularity. The purpose of this paper is to include all the uncertainties that were excluded in the deterministic form (Zeb and Alzahrani, 2021). First, the dynamics of the deterministic model is revised.…”

Section: Introductionmentioning

confidence: 99%

“…First, the dynamics of the deterministic model is revised. The model presented in (Zeb and Alzahrani, 2021) is as follows:…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A stability analysis on a smoking model with stochastic perturbation

Zeb

¹

,

Kumar

²

,

Tesfay

³

et al. 2021

HFF

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Purpose The purpose of this paper is to investigate the effects of irregular unsettling on the smoking model in form of the stochastic model as in the deterministic model these effects are neglected for simplicity. Design/methodology/approach In this research, the authors investigate a stochastic smoking system in which the contact rate is perturbed by Lévy noise to control the trend of smoking. First, present the formulation of the stochastic model and study the dynamics of the deterministic model. Then the global positive solution of the stochastic system is discussed. Further, extinction and the persistence of the proposed system are presented on the base of the reproductive number. Findings The authors discuss the dynamics of the deterministic smoking model form and further present the existence and uniqueness of non-negative global solutions for the stochastic system. Some previous study’s mentioned in the Introduction can be improved with the help of obtaining results, graphically present in this manuscript. In this regard, the authors present the sufficient conditions for the extinction of smoking for reproductive number is less than 1. Research limitations/implications In this work, the authors investigated the dynamic stochastic smoking model with non-Gaussian noise. The authors discussed the dynamics of the deterministic smoking model form and further showed for the stochastic system the existence and uniqueness of the non-negative global solution. Some previous study’s mentioned in the Introduction can be improved with the help of obtained results, clearly shown graphically in this manuscript. In this regard, the authors presented the sufficient conditions for the extinction of smoking, if <1, which can help in the control of smoking. Motivated from this research soon, the authors will extent the results to propose new mathematical models for the smoking epidemic in the form of fractional stochastic modeling. Especially, will investigate the effective strategies for control smoking throughout the world. Originality/value This study is helpful in the control of smoking throughout the world.

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“…Anwar Zeb and Abdullah Alzahrani (Zeb and Alzahrani 2021 ) formulated a mathematical model of smoking. The reversion class was added, and techniques for finding local and global stability related to smoking-free and positive smoking equilibrium points were also given.…”

Section: Introductionmentioning

confidence: 99%

Development of Explicit Schemes for Diffusive SEAIR COVID-19 Epidemic Spreading Model: An Application to Computational Biology

Nawaz

¹

,

Arif

²

,

Ashraf

³

2021

Iran J Sci Technol Trans Sci

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In this contribution, a first-order time scheme is proposed for finding solutions to partial differential equations (PDEs). A mathematical model of the COVID-19 epidemic is modified where the recovery rate of exposed individuals is also considered. The linear stability of the equilibrium states for the modified COVID-19 model is given by finding its Jacobian and applying Routh–Hurwitz criteria on characteristic polynomial. The proposed scheme provides the first-order accuracy in time and second-order accuracy in space. The stability of the proposed scheme is given using the von Neumann stability criterion for standard parabolic PDEs. The consistency for the proposed scheme is also given by expanding the involved terms in it using the Taylor series. The scheme can be used to obtain the condition of getting a positive solution. The stability region of the scheme can be enlarged by choosing suitable values of the contained parameter. Finally, a comparison of the proposed scheme is made with the existing non-standard finite difference method. The results indicate that the non-standard classical technique is incapable of preserving the unique characteristics of the model’s epidemiologically significant solutions, whereas the proposed approaches are capable of doing so. A computational code for the proposed discrete model scheme may be made available to readers upon request for convenience.

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Non-standard finite difference scheme and analysis of smoking model with reversion class

Cited by 13 publications

References 18 publications

A Fractional Model for the Dynamics of Smoking Tobacco Using Caputo–Fabrizio Derivative

A Fractional Model for the Dynamics of Smoking Tobacco Using Caputo–Fabrizio Derivative

A stability analysis on a smoking model with stochastic perturbation

Development of Explicit Schemes for Diffusive SEAIR COVID-19 Epidemic Spreading Model: An Application to Computational Biology

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