Research Article Global Dynamics of a Mathematical Model on Smoking

ISRN Applied Mathematics, Article ID 847075, 7 pages http://dx.doi.org/10.1155/014/847075 Research Article Global Dynamics of a Mathematical Model on Smoking Zainab Alkhudhari, Sarah Al-Sheikh, and Salma Al-Tuwairqi Departmentof Mathematics,KingAbdulaziz University, Jeddah 1551, Saudi Arabia Correspondence should be addressed to Salma Al-Tuwairqi; saltuwairqi@kau.edu.sa Received 17 November 013; Accepted 17 December 013; Published 6 February 014 Academic Editors: A. Bairi and S. W. Gong Copyright 014 Zainab Alkhudhari et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We derive and analyze a mathematical model of smoking in which the population is divided into four classes: potential smokers, smokers, temporary quitters, and permanent quitters. In this model we study the effect of smokers on temporary quitters. Two equilibria of the model are found: one of them is the smoking-free equilibrium and the other corresponds to the presence of smoking. We examine the local and global stability of both equilibria and we support our results by using numerical simulations. 1. Introduction Tobacco epidemic is one of the biggest public health threats the world has ever faced. It kills up to half of its users. Nearly, each year six million people die from smoking of whom more than 5 million are users and ex-users and more than 600,000 are nonsmokers exposed to second-hand smoke. Tobacco users who die prematurely deprive their families of income, raise the cost of health care, and hinder economic development. In smoking death statistics, there is a death caused by tobacco every 8 seconds; 10% of the adult population who smoke die of tobacco related diseases. The World Health Organization predicts that, by 030, 10 million peoplewilldieeveryyearduetotobaccorelatedillnesses.this makes smoking the biggest killer globally. In Saudi Arabia, the prevalence of current smoking ranges from.4 to 5.3% (median = 17.5%) depending on the age group. The results of a Saudi modern study predicted an increase of smokers number in the country to 10 million smokers by 00. The current number of smokers in Saudi Arabia is approximately 6 million, and they spend around 1 billion Saudi Riyal on smoking annually. Clearly smoking is a prevalent problem among Saudis that requires intervention for eradication. Persistent education of the health hazards related to smoking is recommended particularly at early ages in order to prevent initiation of smoking [1, ]. In 1997, Castillo-Garsow et al. [3] proposed a simple mathematical model for giving up smoking. They considered a system described by the simplified PSQ model with a total constant population which is divided into three classes: potential smokers, that is, people who do not smoke yet but mightbecomesmokersinthefuture(p), smokers (S), and people (former smokers) who have quit smoking permanently (Q). Later, this mathematical model was developed by Sharomi and Gumel (008) [4]; they introduced a new class Q t of smokers who temporarily quit smoking and they described the dynamics of smoking by the following four nonlinear differential equations: dp dt = P βps, ds dt = (+γ)s+βps+αq t, (1) dq t = (+α)q dt t +γ(1 σ) S, dq p = Q dt p +σγs. They concluded that the smoking-free equilibrium is globally asymptotically stable whenever a certain threshold, known as thesmokersgenerationnumber,islessthanunityandunstable if this threshold is greater than unity. The public health implication of this result is that the number of smokers in the community will be effectively controlled (or eliminated) atsteadystateifthethresholdismadetobelessthanunity.

ISRN Applied Mathematics Such a control is not feasible if the threshold exceeds unity (a global stability result for the smoking-present equilibrium is provided for a special case). In 011, Lahrouz et al. [5] proved the global stability of the unique smoking-present equilibrium state of the mathematical model developed by Sharomi and Gumel. Also in 011, Zaman [6] derived and analyzed a smoking model taking into account the occasional smokers compartment, and later [7] he extended the model to consider the possibility of quitters becoming smokers again. In 01, Ertürk et al. [8] introduced fractional derivatives into themodelandstudieditnumerically. In this paper, we adopt the model developed and studied in [4, 5] and examine the effect of peer pressure on temporary quitters Q t. By this we mean the effect of smokers on temporary quitters which may be considered as one of the main causes of their relapse. This is done by taking α=αs as a parameter dependent on the pressure of smokers. The aimofthisworkistoanalyzethemodelbyusingstability theory of nonlinear differential equations and supporting the results with numerical simulation. The structure of this paper is organized as follows. In Section, we calculate the smoking generation number, find the equilibria, and investigate their stability. In Section 3, we present numerical simulations to support our results. Finally, we end the paper with a conclusion in Section 4.. Formulation of the Model Let the total population size at time t be denoted by N(t). We divide the populationn(t) into four subclasses: potential smokers (nonsmoker) P(t), smokers S(t), smokers who temporarily quit smoking Q t (t), and smokers who permanently quit smoking Q p (t), suchthatn(t) = P(t) + S(t) + Q t (t) + Q p (t). We describe the dynamics of smoking by the following four nonlinear differential equations: dp dt = P βps, ds dt = (+γ)s+βps+αsq t, dq t = Q dt t αsq t +γ(1 σ) S, dq p = Q dt p +σγs. Here, β is the contact rate between potential smokers and smokers, is the rate of natural death, α is the contact rate between smokers and temporary quitters who revert back to smoking, γ is the rate of quitting smoking, (1 σ)is the fraction of smokers who temporarily quit smoking (at arateγ), and σ is the remaining fraction of smokers who permanently quit smoking (at a rate γ). We assume that the class of potential smokers is increased by the recruitment of individuals at a rate. Inmodel, the total population is constant and P(t), S(t), Q t (t), and Q p (t) are, respectively, the proportions of potential smokers, smokers, temporarily quitters, and permanent quitters where () P(t) + S(t) + Q t (t) + Q p (t) = 1.SincethevariableQ p of model does not appear in the first three equations, we will only consider the subsystem: dp dt = P βps, ds dt = (+γ)s+βps+αsq t, (3) dq t = Q dt t αsq t +γ(1 σ) S. From model 3 we find that dp dt + ds dt + dq t (P+S+Q dt t ). (4) Let N 1 = P + S + Q t ;thenn 1 N 1.Theinitial value problem Φ = Φ,withΦ(0) = N 1 (0), hasthe solution Φ(t) = 1 ce t, and lim t Φ(t) = 1. Therefore, N 1 (t) Φ(t), which implies that lim t sup N 1 (t) 1. Thus, the considered region for model 3 is Γ={(P, S, Q t ) :P+S+Q t 1,P>0,S 0,Q t 0}. (5) All solutions of model 3 are bounded and enter the region Γ. Hence, Γ is positively invariant. That is, every solution of model 3, with initial conditions in Γ, remains there for all t> 0. 3. Equilibria and Smokers Generation Number Setting the right-hand side of the equations in model 3 to zero, we find two equilibria of the model: the smoking-free equilibrium E 0 = (1, 0, 0) and smoking-present equilibrium E =(P,S,Q t ),where P = Q t = and S satisfies the equation +βs, γ (1 σ) S +αs, S +( (β+γβ+α( β)+αγσ) )S βα ( + γσ) (7) + ( + γ β) =0. βα ( + γσ) The smokers generation number R 0 is found by the method of next generation matrix [9, 10]. Let X=(S,Q t,p);thenmodel 3canberewrittenasX = F(X) V(X) such that βps F (X) = [ 0 ], [ 0 ] (8) ( + γ) S αsq t V (X) = [( + α) SQ t γ(1 σ) S]. [ + P + βps ] (6)

ISRN Applied Mathematics 3 By calculating the Jacobian matrices at E 0,wefindthat where D(F (E 0 )) = [ F 0 0 0 ], D(V (E 0)) = [ V 0 J 1 J ], (9) F=[ β 0 ( + γ) 0 ], V=[ ]. (10) 0 0 γ (1 σ) Thus, the next generation matrix is FV 1 =[ β/(+γ) 0 0 0 ] and the smokers generation number R 0 is the spectral radius ρ(fv 1 ) = β/( + γ). Next we explore the existence of positive smoking-present equilibrium E of model 3. Equation (7)hassolutionsofthe form where A 1 and A are S 1, = A 1 ± 1 A 1 4A, (11) A 1 = (β+γβ+α( β)+αγσ), βα ( + γσ) A = ( + γ β) βα ( + γσ) = ( + γ) (1 R 0 ). βα ( + γσ) (1) Proof. Evaluating the Jacobian matrix of model 3 at E 0 (using linearization method [11]) gives β 0 J(E 0 )=[ 0 (+γ)+β 0 ]. (14) [ 0 γ(1 σ) ] Thus, the eigenvalues of J(E 0 ) are λ 1 = λ = < 0, λ 3 = ( + γ) + β. Clearlyλ 3 is negative if R 0 <1.Soall eigenvalues are negative if R 0 < 1, and hence E 0 is locally asymptotically stable. If R 0 =1,thenλ 3 =0and E 0 is locally stable. If R 0 >1,thenλ 3 >0which means that there exists a positive eigenvalue. So E 0 is unstable. Next, we investigate the local stability of the positive equilibrium E by using the following lemma. Lemma 3 (see [1 14]). Let M be a 3 3real matrix. If tr(m), det(m), anddet(m [] ) are all negative, then all of the eigenvalues of M have negative real part. M [] in the previous lemma is known as the second additive compound matrix with the following definition. Definition 4 (see [15] (second additive compound matrix)). Let A = (a ij ) be an n nrealmatrix.thesecondadditive compound of A is the matrix A [] =(b ij ) defined as follows: If R 0 >1,thenA <0and A 1 4A >A 1.Sowehave only one positive solution: S = 1 ( A 1 + A 1 4A ). (13) n=:a [] =a 11 +a, a 11 +a a 3 a 13 n=3:a [] = a 3 a 11 +a 33 a 1. a 31 a 1 a +a 33 (15) If R 0 =1,thenA =0and there are no positive solutions. If R 0 <1and β,thena 1 >0and A >0and there are no positive solutions. These results are summarized below. Theorem 1. Model 3 always has the smoking-free equilibrium point E 0 = (1, 0, 0). As for the existence of a smoking-present equilibrium point E, one has three cases: (i) if R 0 <1and β, one has no positive equilibrium point; (ii) if R 0 >1, one has one positive equilibrium point E ; (iii) if R 0 =1, one has no positive equilibrium point. 4. Stability of the Equilibria 4.1. Local Stability. First, we investigate the local stability of E 0 and state the following theorem. Theorem (local stability of E 0 ). If R 0 <1,thesmoking-free equilibrium point E 0 of model 3 is locally asymptotically stable. If R 0 =1, E 0 is locally stable. If R 0 >1, E 0 is unstable. Theorem 5 (local stability of E ). The smoking-present equilibrium E of model 3 is locally asymptotically stable if α β. Proof. Linearizing model 3 at the equilibrium E (P,S,Q t ) gives βs βp 0 J(E )= βs 0 αs. (16) [ 0 αq ] t +γ(1 σ) αs [ ] The second additive compound matrix J [] (E ) is J [] (E ) βs αs 0 = αq t +γ(1 σ) βs αs βp. [ 0 βs αs ] [ ] (17)

4 ISRN Applied Mathematics Here tr (J (E )) = βs αs <0, det (J (E )) = βαs (Q t βp S )+(αq t β P S ) βαs (Q t βp S ) +β(q t βp S ), if α β =(βαs +β)(q t +P 1)<0, since P +Q t <1, det (J [] (E )) = ( + βs )[(+βs +αs )(+αs ) +β P S ]+αq t (+αs ) = (+βs )[(+βs +αs )(+αs )] +(αq t β P S ) +S (α Q t β3 P S )<0, if α β. (18) Hence, by previous lemma, E is locally asymptotically stable if α β. 4.. Global Stability. We will examine the global stability of E 0 when β.first,itshouldbenotedthatp<1in Γ for all t>0. Consider the following Lyapunov function [16]: L=S+Q t, dl dt = (+γ)s+βps+αsq t Q t αsq t +γ(1 σ) S ( +β)s Q t 0. (19) We have dl/dt < 0 for β and dl/dt = 0 only if S=0and Q t =0. Therefore, by LaSalle s Invariance Principle [16], every solution to the equations of the model with initial conditions in Γ approaches E 0 as t.wehavethefollowingresult. Theorem 6 (global stability of E 0 ). If β,thene 0 is globally asymptotically stable in Γ. Now we examine the global stability of E by using the following theorem [1, 17] to prove that model 3 has no periodic solutions, homoclinic loops, and oriented phase polygons inside the invariant region. Theorem 7. Let g(p, S, Q t ) = {g 1 (P, S, Q t ), g (P,S,Q t ), g 3 (P, S, Q t )} be a vector field which is piecewise smooth on Γ and which satisfies the conditions g f = 0, (curl g) n<0 in the interior of Γ,wheren is the normal vector to Γ and f=(f 1,f,f 3 ) is a Lipschitz continuous field in the interior of Γ and curl g=( g 3 / S g / Q t ) i ( g 3 / P g 1 / Q t ) j+ ( g / P g 1 / S) k. Then the differential equation system P = f 1, S = f,andq t = f 3 has no periodic solutions, homoclinic loops, and oriented phase polygons inside Γ. Let Γ ={(P,S,Q t ) : P+((+γσ)/)S+Q t =1,P>0,S 0, Q t 0}.OnecaneasilyprovethatΓ Γ, Γ is positively invariant, and E Γ.Thus,westatethefollowingtheorem. Theorem 8. Model 3 has no periodic solutions, homoclinic loops, and oriented phase polygons inside the invariant region Γ. Proof. Let f 1, f, and f 3 denote the right-hand side of equations in model 3, respectively. Using P + (( + γσ)/)s + Q t =1to rewrite f 1, f,andf 3 in the equivalent forms, we have f 1 (P, S) = P βps, f 1 (P, Q t )= P βp[ +γσ (1 P Q t)], f (P, S) = (+γ)s+βps+αs[1 P ( +γσ )S], f (S, Q t )= (+γ)s +β[1 Q t ( +γσ )S]S+αSQ t, f 3 (P, Q t )= Q t [αq t γ(1 σ)][ +γσ (1 P Q t)], f 3 (S, Q t )= Q t αsq t +γ(1 σ) S. Let g=(g 1,g,g 3 ) be a vector field such that g 1 = f 3 (P, Q t ) PQ t α = P(+γσ) + + f (P, S) PS α +γσ + αq t P(+γσ) γ (1 σ) γ (1 σ) PQ t ( + γσ) Q t ( + γσ) γ (1 σ) P(+γσ) + γ P β α P +α(+γσ ) S P +α, g = f 1 (P, S) PS = PS f 3 (S, Q t ) SQ t β+α γ (1 σ) Q t, (0)

ISRN Applied Mathematics 5 Potential smokers P(t) 1.0 0.9 0.8 0.7 0.6 (a) Smokers S(t) 0.5 0.0 0.00 (b) 0.14 Temporary quitters Q t (t) Permanent quitters Q p (t) 0.1 0.08 0.06 0.04 0.0 0.00 0.00 1 3 4 1 3 4 (c) (d) Figure 1: Time plots of model with different initial conditions for R 0 <1. (a) Potential smokers; (b) smokers; (c) temporary quitters; (d) permanent quitters. g 3 = f (S, Q t ) SQ t f 1 (P, Q t ) PQ t ( + γ) = + β β β(+γσ)s Q t Q t Q t +α + β + PQ t Q t Q t ( + γσ) βp Q t ( + γσ) β ( + γσ). (1) Since the alternate forms of f 1, f,andf 3 are equivalent in Γ, then Using the normal vector n = (1, ( + γσ)/, 1) to Γ,we can see that (curl g) n= βγσ γ (1 σ) Q t PQt +γσ P Q t P S αγσ P <0. (3) Hence, model 3 has no periodic solutions, homoclinic loops, and oriented phase polygons inside the invariant region Γ. Consequently,wehavethefollowingresult. So g f=0on Γ. g f=g 1 f 1 +g f +g 3 f 3 = f 3f 1 PQ t f f 1 PS + f 1f PS f 3f SQ t + f f 3 SQ t f 1f 3 PQ t =0. () Theorem 9 (global stability of E ). If R 0 > 1, then the smoking-present equilibrium point E of system (3) is globally asymptotically stable. Proof. We know that, if R 0 > 1 in Γ,thenE 0 is unstable. Also Γ is a positively invariant subset of Γ and the ω limit set of each solution of (3) is a single point in Γ since there is no periodic solutions, homoclinic loops, and oriented phase

6 ISRN Applied Mathematics Temporary quitters Q t (t) Potential smokers P(t) 0.8 0.7 0.6 0.5 0.4 0.5 0.0 (a) Smokers S(t) 0.5 0.0 (b) 0.5 Permanent quitters Q p (t) 0.0 1 3 4 1 3 4 (c) (d) Figure : Time plots of model with different initial conditions for β α. (a) Potential smokers; (b) smokers; (c) temporary quitters; (d) permanent quitters. polygons inside Γ. Therefore E is globally asymptotically stable. 5. Numerical Simulations In this section, we illustrate some numerical solutions of model for different values of the parameters, and we show that these solutions are in agreement with the qualitative behavior of the solutions. We use the following parameters: = 0.04, γ = 0.3, α = 0.5, andσ = 0.4, withtwo different values of the contact rate between potential smokers and smokers: for R 0 <1we use β = 0. and for β αwe use β = 0.5. Model is simulated for the following different initial values such that P+S+Q t +Q p =1: (1) P(0) = 0.80301, S(0) = 68, Q t (0) = 0.0860,and Q p (0) = 0.00811; () P(0) = 0.75000, S(0) = 0.1677, Q t (0) = 0.07000,and Q p (0) = 0.018; (3) P(0) = 0.70000, S(0) = 0.1800, Q t (0) = 566,and Q p (0) = 0.0634; (4) P(0) = 0.63400, S(0) = 0.8800, Q t (0) = 0.04800,and Q p (0) = 0.03000. For R 0 = 0.58835 < 1, Figure 1(a) shows that the number of potential smokers increases and approaches the total population 1. Figure 1(b) shows that the number of the smokers decreases and approaches zero. In Figures 1(c) and 1(d), temporary quitters and permanent quitters increase at first; after that they decrease and approach zero. We see from Figure 1 that, for any initial value, the solution curves tend to the equilibrium E 0,whenR 0 < 1.Hence,modelis locally asymptotically stable about E 0 for the previous set of parameters. In Figure, we use the same parameters and initial values as previously with β = 0.5. Figure (a) shows that the number of potential smokers decreases at first; then it increases and approaches P. Figure (b) shows that the number of smokers increases at first; then it decreases and approaches S.In Figures (c) and (d), temporary quitters and permanent quitters increase at first; after that they decrease and approach Q t and Q p.weseefromfigure that, for any initial value, thesolutioncurvestendtotheequilibriume,whenβ α.

ISRN Applied Mathematics 7 Hence, model is locally asymptotically stable about E for the above set of parameters. 6. Discussion and Conclusions In this paper, we presented a mathematical model to analyze the behavior of smoking dynamics in a population with peer pressure effect on temporary quitters Q t.localasymptotic stability for the smoking-free equilibrium state is obtained when the threshold quantity R 0 is less than 1 (i.e., when the contact rate β between potential smokers and smokers is less than the sum of natural death rate and the quitting rate γ). A Lyapunov function is used to show global stability of thesmoking-freeequilibriumwhenthecontactratebetween potential smokers and smokers is less than or equal to the natural death rate (β ). This means that the number of smokers may be controlled by reducing the contact rate β to be less than the natural death rate. On the other hand, if β α (i.e., when the contact rate β between potential smokers and smokers is greater than the contact rate α between smokers andtemporaryquitterswhorevertbacktosmoking),thenthe smoking-present equilibrium state is locally asymptotically stable. By showing that this model has no periodic solutions, homoclinic loops, and oriented phase polygons inside the invariant region Γ,weprovedtheglobalasymptoticstability of E. This means that, if R 0 >1, smoking will persist. Some numerical simulations are performed to illustrate the findings of the analytical results. For R 0 >1, the results show that the smokers population reaches a steady state of approximately 6% of the total population. This model may be extended to include, for example, smokers who after quitting smoking may become potential smokersagain.wemayalsoconsiderthepublichealthimpact ofsmokingrelatedillnessesaswell. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. [6] G. Zaman, Qualitative behavior of giving up smoking models, Bulletin of the Malaysian Mathematical Sciences Society,vol.34, no., pp. 403 415, 011. [7] G. Zaman, Optimal campaign in the smoking dynamics, Computational and Mathematical Methods in Medicine, vol. 011, Article ID 163834, 9 pages, 011. [8] V. S. Ertürk, G. Zaman, and S. Momani, A numeric-analytic method for approximating a giving up smoking model containing fractional derivatives, Computers & Mathematics with Applications,vol.64,no.10,pp.3065 3074,01. [9] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences,vol.180, pp.9 48,00. [10] F. Brauer, P. van den Driessche, and J. Wu, Mathematical Epidemiology, vol. 1945 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 008. [11] L. Perko, Differential Equations and Dynamical Systems, vol. 7 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1991. [1]L.Cai,X.Li,M.Ghosh,andB.Guo, Stabilityanalysisof an HIV/AIDS epidemic model with treatment, Computational and Applied Mathematics,vol.9,no.1,pp.313 33, 009. [13] C. C. McCluskey and P. van den Driessche, Global analysis of two tuberculosis models, Dynamics and Differential Equations,vol.16,no.1,pp.139 166,004. [14] J. Arino, C. C. McCluskey, and P. van den Driessche, Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM Journal on Applied Mathematics, vol.64,no.1,pp.60 76,003. [15] X. Ge and M. Arcak, A sufficient condition for additive Dstability and application to reaction-diffusion models, Systems & Control Letters, vol. 58, no. 10-11, pp. 736 741, 009. [16] M. W. Hirsch, S. Smale, and R. L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, vol. 60 of Pure and Applied Mathematics, Elsevier/Academic Press, Amsterdam, The Netherlands, nd edition, 004. [17] S. Busenberg and P. van den Driessche, Analysis of a disease transmission model in a population with varying size, Journal of Mathematical Biology,vol.8,no.3,pp.57 70,1990. References [1] M. M. Bassiony, Smoking in Saudi Arabia, Saudi Medical Journal,vol.30,no.7,pp.876 881,009. [] World Health Organization report on tobacco, 01, http://www.who.int/mediacentre/factsheets/fs339/en/. [3] C. Castillo-Garsow, G. Jordan-Salivia, and A. R. Herrera, Mathematical models for the dynamics of tobacco use, recovery, and relapse, Technical Report Series BU-1505-M, Cornell University, Ithaca, NY, USA, 1997. [4] O. Sharomi and A. B. Gumel, Curtailing smoking dynamics: a mathematical modeling approach, Applied Mathematics and Computation,vol.195,no.,pp.475 499,008. [5] A. Lahrouz, L. Omari, D. Kiouach, anda. Belmaâti, Deterministic and stochastic stability of a mathematical model of smoking, Statistics & Probability Letters, vol. 81, no. 8, pp. 176 184, 011.

Advances in Operations Research Advances in Decision Sciences Mathematical Problems in Engineering Algebra Probability and Statistics The Scientific World Journal International Differential Equations Submit your manuscripts at International Advances in Combinatorics Mathematical Physics Complex Analysis International Mathematics and Mathematical Sciences Stochastic Analysis Abstract and Applied Analysis International Mathematics Discrete Dynamics in Nature and Society Discrete Mathematics Applied Mathematics Function Spaces Optimization