Fractional Order Model on Giving up Smoking

Fractional Order Model on Giving up Smoking

A. George Maria Selvam1,

1 Sacred Heart College, Tirupattur – 635 601, S.India

R. Janagaraj2

2Kongunadu College of Engineering andTechnology, Thottiam-621 215, S.India.

D. Abraham Vianny3

3Knowledge Institute of Technology, Kakapalayam-637 504, S.India

Abstract – In this paper, we analyze a fractional order model on

Definition 2.[5] The Caputo definition is

smoking in which the population is divided into three classes: potential smokers, smokers and quitters. Using fractional order differential dynamical theory, we study the effect of smokers on quitters. The equilibrium points are established and stability of

() =

1 [ ]

( )

< )

()()

( )+1

, ( 1

the equilibrium points is discussed. Finally we examine the stability of both equilibriums and the results are illustrated by numerical simulations, and they exhibit rich dynamics of the fractional model.

Keywords: Fractional Order, Differential equations, Smoking, Equilibrium points, stability.

The stability results for the fractional order linear system are given below.

Lemma 3. [6, 7] The fractional – order autonomous system

= , (0) = 0

where 0< < 1 , and is

1. asymptotically stable if and only if

   | | ( )

   1. INTRODUCTION

      Tobacco epidemic threatens the lives of one billion men,

      arg(())

2. stable if and only if

> 2 ,

= 1,2, , .

women and children during this century. Tobacco use can kill in so many ways that it is a risk factor for six of the eight leading causes of death in the world. The cure for this devastating epidemic is dependent not on medicines or vaccines, but on the concerted actions of government and civil society. Research on health effects of tobacco has focused primarily on cigarette tobacco smoking. In 1950, Richard Doll published research findings in the British Medical Journal showing a close link between smoking and lung cancer. Four years later, in 1954, a study by some 40,000 doctors over 20 years, confirmed the suggestion, based on which the government issued advice that smoking and lung cancer rates were related. An estimated 5.4 million people die manually from lung cancer, heart disease and other illnesses due to tobacco. If Unchecked, the number may increase to more than 8 million a year by 2030.

1. REVIEW OF FRACTIONAL CALCULUS Fractional order calculus deals with integrals and derivatives of arbitrary order. Fractional calculus is a natural extension of classical calculus. The origin of the theory of fractional calculus can be traced back to a letter dated September 30th, 1695 written by L'Hopital to Leibniz. Fractional calculus has emerged as one of the most important interdisciplinary subjects in Mathematics, Physics, Biology and Engineering. Definition 1. [5] Riemann-Liouville definition is

   |arg( ())| , ( = 1,2, , ).

   2

   wherearg(()) denotes the argument of the eigenvalue

   of .

2. MODEL DESCRIPTION OF FRACTIONAL ORDER In 1997, Castillo-Garsow et al. [4] proposed a

   system described by model for giving up smoking. The total population ()is divided into three classes: potential smokers () , smokers () , and people who have quit smoking permanently (), such that() = () + () +

   (). The mathematical model of smoking is described by the following three nonlinear differential equations: [9]

   = () ()()

   = ( + )() + ()()

   + () (1)

   = ( + )() + (1 ) ()

   In (1), is the contact rate between potential smokers and

   smokers, is the rate of natural death, is the contact rate between smokers and quitters who revert back to smoking, is the rate of quitting smoking, (1 ) is the fraction of smokers who quit smoking (at a rate ).

   Several authors formulated fractional order systems and

   () =

   1 [ ]

   ()

   , ( 1

   analyzed the dynamical and qualitative behavior of the systems [1, 3, 8]. Following this trend, In this paper, we

   ( )

   < )

   ( )+1

   propose a system of fractional order smoking model. We

   assume the following fractional order Mathematical Model on Smoking.

   1 () = () ()()

   2 () = ( + )() + ()()

   + () (2)

   3 () = ( + )() + (1 ) ()

   where the parameter , , , and are all positive and

   1 , 2 , 3 are all fractional derivative orders.

3. STABILITY ANALYSIS OF EQUILIBRIUM POINTS To evaluate the equilibrium point, let us consider

   1 () = 0; 2 () = 0; 3 () = 0.

   The fractional order system has two equilibrium points; the smoking-free equilibrium 0 = (1, 0, 0) and the smoking-

   If 12 > 3 , the Routh Hurwitz criterion implies that all roots of (1)() have negative real parts, 1 is stable. This conditions are in contrast to the existence condition of 1. It means that1 is unstable.

4. NUMERICAL SOLUTIONS AND EXAMPLES Numerical solution of the fractional- order smoking model is given as follows [7]:

   () = ( (1) (1)(1))1

   (1)()

   =

   () = (( + )(1) + (1) (1)

   present equilibrium 1 = (, , ), where

   + ( ))2 (2)( )

   =

   ( + )

   + ( + ) ,

   1

   =

   ( + )

   = ( + ) + ( + )

   ,

   () = (( + )(1) + (1 ) (1)) 3

   (3)()

   ( 1) ( 1)

   = ( + ) ( + ) + ( + ) .

   =

   Here 0 = ( +)

   . Following lemma [2] is needed

   where is the simulation time, = 1,2,3, , , for =

   (+)+ ( +)

   for the analysis of the stability properties.

   [

   ] , and ((0), (0), (0)) is the start point (initial

   Lemma 4.If 0 < 1 , the smoking-free equilibrium 0 is asymptotically stable. If 0 = 1 , 0 is Stable; 0 > 1 , 0 is Unstable.

   Based on (2), to investigate the stability of each equilibrium point (, , ), we provide the Jacobian matrix

   (, , ).

   0

   conditions). In this section we develop several numerical simulations of the fractional order model (2).

   Example 1. Let us consider the parameter values = 0.5, = 0.05, = 0.5, = 0.9 and = 0.4 with the initial conditions (0) = 0.65, (0) = 0.30, (0) = 0.05. Also we take the fractional derivatives 1 = 2 = 3 = 0.85. The eigen values are 1 = 0.90, 2 = 0.9869 and 3 =

   For

   = [ ( + ) + ].(3) 0 (1 ) ( + )

   , we have

   1.7631 . The time plot of potential smokers(nonsmokers)

   (), smokers () and quit smoking () diagram illustrate the result, see Figure-1. Here reproduction number 0 =

   0

   0

   ( +)

   (+)+ ( +)

   = 0.07 = 0.0386 < 1 and for the fractional

   1.81

   (0) = [ 0 ( + ) + ] 0 (1 ) ( + )

   order system(2), |arg(1,2,3)| = 3.1416 > 1.3357 =

   . Hence Lemma (4) and Lemma (3), the smoking-free

   Trace ( ) = [3 + + ] and Det ( ) = 2

   0 0 equiibrium 0 of the system (2) isasymptotically stable and

   2

   2[ ] + [ ] . The eigen values of matrix (0) are 1 = and 2,3 = ±

   the characteristic equation of the linearized system(2) at the smoking-free equilibrium 0 is ([7])

   1 ( + )2 + [2( ) + ] 4 .

   2

   while for 1we have

   255

   + 2.65170

   + 3.31585

   + 1.566 = 0

   0

   (1) = [ ]

   0 (1 ) ( + )

   =

   where ( +)

   (+)+ ( + )

   and = ( + )

   ( + )

   . The

   characteristic polynomial(1)() for (1) is

   1

   ( )() = 3 + 12 + 2 + 3

   where

   1 = [ + + ] [ ],

   2 = ( + )[ + ] (1 ) + ( + )

   ( + ),

   3 = ( + )[( + ) ( + )] (1 ).

   Figure.1 Time series of smoking-free equilibrium 0 with Stability of 0 < 1.

   Example 2. Let us consider the parameter values = 0.25, = 0.15, = 0.02, = 0.04 and = 0.3 with the initial conditions (0) = 0.65, (0) = 0.30, (0) = 0.05 . Also we take the fractional derivatives 1 = 2 = 3 = 0.85, it is the smoking-present equilibrium with the approximate solutions ((), (), ()) = (0.319, 0.567, 0.027). The eigen values are 1 =

   0.0567, 2 = 0.0693 and 3 = 0.3012 . The time plot of potential smokers(nonsmokers) () , smokers () and quit smoking () diagram illustrate the result, see Figure-2.

   255 + 0.4272170 + 0.041985 + 0.0012 = 0.

   0 =

   Here reproduction number ( +)

   (+)+ ( +)

   = 0.0435 =

   0.0139

   2

   1. > 1 and for the fractional order autonomous system(2), |arg(1,2,3)| = 3.1416 > 1.3357 = . Hence Lemma (4) and Lemma (3), the smoking-present equilibrium

      1 of the system (2) is stable and the characteristic equation

      of the linearized system(2) at the smoking-present equilibrium

      1 is ([7])

      Figure.2 Time series of smoking-present equilibrium 1 with Stability of

      0 > 1.

5. CONCLUSION

In this paper, we presented the Fractional Order Model on giving up smoking. Dynamic properties are discussed by computing equilibrium points and their stability properties. Finally numerical examples are presented for various fractional orders.

Figure.3 Time series of smoking-present equilibrium 1and different fractional derivatives () with Stability.

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