# Approximate Solution and Analysis of Smoking Epidemic Model with Caputo Fractional Derivatives

• M. Abdullah
• Nauman Raza
• M. Farman
• M. O. Ahmad
Original Paper

## Abstract

In this paper, an epidemic model is presented to describe the dynamics of drugs usage among the adults. The Caputo fractional derivative operator of order $$\phi \in (0, 1]$$ is employed to obtain the system of fractional differential equations. Smoking is the large problem in the entire world. Despite the overwhelming facts about smoking, it is still a bad habit which is widely spread and socially accepted. A non-linear mathematical model is established to study and assess the dynamics of smoking and its impact on public health in a community. The analysis of two different states, disease free and endemic which means the disease dies out or persist in a population has been made by presenting sensitivity analysis. The stability of the model is provided by threshold or reproductive number. The Laplace Adomian decomposition method has been employed to solve smoking model which has five compartments, potential smoker P, occasional smokers O, smokers S, temporarily quit smokers Q and permanently quit smokers L. Finally, some numerical results are presented which shows the effect of fractional parameter $$\phi$$ on our obtained solutions.

## Keywords

Epidemic model Endemic equilibrium Qualitative analysis Stability analysis Fractional order

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© Springer Nature India Private Limited 2018

## Authors and Affiliations

• M. Abdullah
• 1
• 2
• Nauman Raza
• 3
• M. Farman
• 2
• M. O. Ahmad
• 2
1. 1.Department of MathematicsUniversity of Engineering and TechnologyLahorePakistan
2. 2.Department of Mathematics and StatisticsThe University of LahoreLahorePakistan
3. 3.Department of MathematicsUniversity of the PunjabLahorePakistan