Dynamical transmission and effect of smoking in society

Article history: Received 27 September 2017 Received in revised form 6 December 2017 Accepted 10 December 2017 Smoking is a large problem in the entire world. Despite overwhelming facts about smoking, it is still a very bad habit which is widely spread and accepted socially. Among smokers, often the desire to quit smoking arises. A large number of smokers attempt to quit, but only a few of them are successful. In this research, the nonstandard finite difference scheme is applied on system which is dynamically consistent, easy to implement and show a good agreement to control the bad impact of smoking for long period of time and to eradicate a death killer factor in the world spread by smoking. We have discussed the qualitative behavior of the model and numerical simulations are carried out to support the analytical results.


Introduction
*The scope of mathematics includes mathematical modeling and esoteric mathematics. The flow of work, process, predictions and outcomes can easily be measured with the help of mathematical concepts and theory. Therefore, biologists are now extremely dependent on mathematics. Mathematical modeling of biological sciences is done by many brilliant scientists (Biazar, 2006;Busenberg and Van den Driessche, 1990;El-Sayed et al., 2009). The relationship between simple mathematical modeling involves biological system, integer order differential equations that show their dynamics and complex system which describes their changing of structure. The nonlinearity and multi-scale behaviors in mathematical modeling describe the mutual relationship between parameters (Makinde, 2007). In last few decades, many biological models were studied in detail by using classical derivatives (Arafa et al., 2012;Kribs-Zaleta, 1999;Buonomo and Lacitignola, 2008;Liu and Wang, 2010;Haq et al, 2017).
Smoking is the major problem in the entire world effecting healthy community. Smoking effects different organs of human body caused more than one million deaths in the world. A chance of heart attack in smoker is 70% more as compared to nonsmoker. Similarly the incident rate of lung cancer of smoker is 10% more than nonsmoker. The main effects of short term smoking are coughing, stained teeth, high blood pressure and bad breath. The major effects of long term smoking are gum disease, stomach ulcer, lung cancer, heart disease, throat cancer and mouth cancer in the recent years. The life of smoker is also 12 to 13 years shorter than nonsmoker. According to the reports of WHO smoking kills many individual in the entire world. Every scientist, doctor and mathematician tries to control the effect of smoking and mathematician tried for the formation of different valuable smoking models to overcome the smoking effects. The lot of smoking models was planned by the authors. Examined the smoking model related with Caputo fractional derivative (Erturk et al., 2012). Examined the optimal control of the smoking models and present the qualitative analysis of dynamics of smoking (Zaman, 2011a;2011b). Analysis of cigarette smoking and lung cancer (Lubin and Caporaso, 2006). Describe the mathematical analysis of the dynamics of tobacco us their recovery and decline (Garsow et al., 2000). Explained the smoking fractional mathematical model (Mickens, 1989). Established the curtailing smoking dynamics (Sharomi and Gumel, 2008). Examined a fractional smoking and many others (Zeb et al., 2012). Interpret the description of smoking global dynamics of mathematical system of equations (Alkhudhari et al., 2014).
In order to overcome the numerical instabilities phenomena in 1989 Mickens introduced the concept nonstandard finite difference scheme and after that has developed NSFD methods in many works, such as (Mickens, 1994;2000;2002;2005). According to Mickens, NSFD schemes are those constructed following a set of five basic rules. The NSFD schemes preserve main properties of the differential counterparts, such as positivity, monotonicity, periodicity, stability, and some other invariant including energy and geometrical shapes. It should be emphasized that NSFD schemes can preserve all properties of continuous models for any discretization parameters. The discrete models with these properties are called dynamically consistent (Anguelov et al., 2014).
In this paper, the total population is divided into five compartments potential smokers, occasional smokers, heavy smokers, temporary quitters and permanently quitters. We also investigate the stability and qualitative analysis of smoking model. An unconditionally convergent nonstandard finite difference scheme has been presented to obtain solution of model. The analysis of disease free equilibrium has been made by finding reproductive number. Numerical results are presented graphically to show the dynamics of the model.

Mathematical model
Smoking model is divided into five sub compartment like potential smokers ( ), occasional smokers ( ), heavy smokers ( ) temporary quitters ( ) and smokers who quit permanently ( ) specified by ( ) = ( ) + ( ) + ( ) + ( ) + ( ). The proposed smoking model (Singh et al., 2017) in the form of system of nonlinear differential equation is given as: Here indicates the contact rate between potential smokers and smokers who smoke occasionally, represents the contact rate between temporary quitters and smokers who smoke occasionally, represents the rate of giving up smoking, (1 − ) stances for the fraction of smokers who temporarily give up smoking at a rate , indicate the contact rate between smokers and temporary quitters who return back to smoking, denotes the rate of natural death, denotes the remaining fraction of smokers who give up smoking permanently (at a rate ).

Disease free equilibrium
By substituting the values of parameters in given system of differential equations and the rate of change with respect to time is zero, we get.

Stability and sensitivity analysis
It is important to find the verge conditions to check the status of population, whether the disease persist or dies out. In case of disease free equilibrium point, 0 < 1, which shows that the disease will dies out. In case of endemic equilibrium, 0 > 1, which shows that the disease spreads in the population, where 0 is the reproductive number is also known basic reproductive ratio or basic reproductive rate. Consider the jacobian matrix as Theorem 1: The disease free equilibrium 0 is locally asymptotically stable for 0 < 1, otherwise unstable. [−2 − − + √ 2 + 2 − 4 + 2 ] < 0.
All the Eigen values of the above matrix according to the disease free equilibrium point are negative real parts which represents that the given system is locally asymptotically stable.
Sensitivity of 0 can be analyzed by taking the partial derivatives of reproductive number for the involved parameters as follows = − (1− )(2 + + ) It can be seen that 0 is most sensitive to change in parameter, here, 0 is increasing with , and decreasing with , . In other words it found that the sensitivity analysis shows that prevention is better than to control the disease.
Note that the denominator function ∅ is expected to better capture the dynamics of the continuous model through the presence of the underlying parameters , . In fact, exact schemes for a wide range of dynamical systems involve such complex denominator functions (Lubuma and Patidar, 2007;Gumel, 2014).

Results and discussion
The mathematical analysis of epidemic smoking model with non-linear incidence has been presented. To observe the effects of the parameters using in this dynamics smoking model (1)-(5), conclude several numerical simulations varying the value of parameters for 0 < 1. Figs. 1-5 show the convergence solution for diseases free equilibrium by using NSFD scheme at ℎ = 1, ℎ = 5 and ℎ = 10 for ∅ = ∅(ℎ) + (ℎ 2 ). The technique create a better impact to control the smoking, it reduces the infected rate and increases the potential smokers during disease Free State. In Fig. 1 by decreasing the value of ℎ potential smokers( )increases with time. In Fig. 2 by decreasing the value of ℎ occasional smokers ( ) decreases rapidly with time. In Fig. 3 by decreasing the value of ℎ heavy smokers ( ) decreases rapidly with time. In Fig. 4, by decreasing the value of ℎ temporary quitters ( ) increases initially, but after some time decreases rapidly with time. In Fig. 5 by decreasing the value of ℎ permanent quitters ( ) increases initially, but after some time decreases rapidly with time. It can be easily seen that by reducing the step size the system (1)-(5) converge rapidly to the steady state point.

Conclusion
Sufficient conditions for local stability of the DFE point 0 are given by using the basic reproduction number 0 of the model, where it is asymptotically stable and sensitivity analysis of the parameters involved in threshold parameter 0 , which shows the actual behavior of the dynamical model to reduce the smoking effect in the society. It is important to note that nonstandard finite difference scheme for mathematical models based on system of differential equations is more powerful approach to compute the convergent solutions for the disease models. Finally, we presented the numerical simulation and verified all the analytical results numerically by using nonstandard finite difference scheme to reduce the infected rates very fast for disease free equilibria by using different step size, we are able to control the spreading of smoking in the community.