Comp. Appl. Math.DOI 10.1007/s40314-013-0054-2

Application of series method with Padé and Laplace-Padéresummation methods to solve a model for the evolutionof smoking habit in Spain

Hector Vazquez-Leal · Francisco Guerrero

Received: 18 February 2013 / Revised: 27 May 2013 / Accepted: 8 June 2013© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013

Abstract We obtain approximated analytical solutions of a mathematical model of the evo-lution of smoking habit in Spain (Guerrero et al. in Int J Drug Policy 22:247–251, 2011) usingthe series method. To enlarge the domain of convergence, we apply the Padé and Laplace-Padé resummation methods to the series solution. We present a comparison of our resultsand a solution obtained using homotopy analysis method (Guerrero et al. in Nonlinear AnalReal World Appl 14:549–558, 2013b), resulting that the combination of series method withLaplace-Padé resummation method generates the best results for the complete domain.

Keywords Smoking habit · Series method · Padé · Laplace-Padé

Mathematics Subject Classification (2000) 34L30

1 Introduction

The fact that epidemic models consist of a system of non-linear differential equations under-lines the importance of having reliable methods for solving them. This type of models can be

H. Vazquez-Leal gratefully acknowledge the financial support provided by the National Council for Scienceand Technology of Mexico (CONACyT) through grant CB-2010-01 #157024 and PROMEP-Red:“Instrumentacin-de-Sensores-para-aplicaciones-de-Fisiologa-y-Biomedicina”. In addition, the first authorwould like to thank Rogelio-Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their contribution tothis work. F. Guerrero would like to thank the financial support by the Universitat de València GrantUV-INV-PRECOMP12-80708.

H. Vazquez-Leal (B)Electronic Instrumentation and Atmospheric Sciences School,University of Veracruz, Cto. Gonzalo Aguirre Beltrán S/N,91000 Xalapa, Veracruz, Méxicoe-mail: hvazquez@uv.mx

F. GuerreroDepartment of Applied Mathematics, University of Valencia,Calle Dr. Moliner 100, Burjassot, 46100 Valencia, Spaine-mail: guecor@uv.es

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H. Vazquez-Leal, F. Guerrero

integrated using any standard numerical method. However, it is known that these algorithmscan give some problems, such as numerical instabilities, oscillations or false equilibriumstates, among others. This means that the numerical solution may not correspond to the realsolution of the original system of differential equations (Lambert 1991). This is the reasonwhy we are interested in obtaining a continuous solution in the form of an analytical approx-imation to the real solution. Among the different methods that have been developed to obtainanalytical approximations for the solution of a system of ODEs, we can use the series method(SM) (Forsyth 1906; Geddes 1979; Ince 1956), homotopy perturbation method (Filobello-Nino et al. 2012a,b; He 2007, 2012; He and Wu 2007; Khan et al. 2011, 2012a,b; Koak etal. 2011; Vazquez-Leal 2012; Vazquez-Leal et al. 2012a,b,c,d), homotopy analysis method(Guerrero et al. 2013b; Yue Tan and Saeid Abbasbandy 2008), among others.

The epidemic model for smoking habit in Spain is a system of non-linear differentialequations without closed solution (Guerrero et al. 2011). The interest of this model is that ithas been able to describe correctly the real evolution of the spread of the smoking habit inSpain. It was constructed using real data for the initial values and for the parameters of thesystem. Constant population is assumed by taking birth and death rates equal and differentfrom zero.

In this work, we propose the use of the series method to obtain an analytical approximationto the solution smoking habit model. Moreover, in order to enlarge the domain of the powerseries, we propose the use of Padé and Laplace-Padé resummation methods. The proposedsolutions are compared with a HAM solution reported in Guerrero et al. (2013b), resultingthat the solution obtained with the series method and treated with Laplace-Padé methodgenerates the most accurate reported approximated solution valid for t = [0,∞] years.

This paper is organized as follows. In Sect. 2, we provide a brief review of the seriesmethod. Section 3 presents the basic concept of Padé and Laplace-Padé resummation meth-ods. In Sect. 4, we introduce the mathematical model of the evolution of smoking habit inSpain. After that, we present the solution obtained with the series method for the smokinghabit model in Sect. 5. In Sect. 6, we present the resulting power series solution. Next, Sect. 7shows results and discuss our findings. Finally, a concluding remark is given in Sect. 8.

2 Basic concept of series method

It can be considered that a nonlinear differential equation can be expressed as

A(u) − f (t) = 0, (1)

where A is a general operator and f (t) is a known analytic function of independent variablet .

The series method establishes that the solution of a differential equation can be written as

u =∞∑

i=0

ui ti , (2)

where u0, u1, . . . are unknowns to be determined by series method.The basic process of series method can be described as:

(1) Equation (2) is substituted into (1), then we regroup equation in terms of t-powers.(2) We equate each coefficient of the resulting polynomial to zero.(3) The boundary conditions of (1) are substituted into (2) to obtain an approximation for

each initial condition.

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(4) Afore-mentioned steps generate a nonlinear algebraic equation system (NAEs) in termsof the unknowns of (2).

(5) Finally, we solve the NAEs to obtain u0, u1, . . ., coefficients.

3 Padé and Laplace-Padé resummation methods

Several approximated methods provide power series solutions (polynomial). Nevertheless,sometimes, this type of solutions lacks large domains of convergence. Therefore, Padé (Barar-nia et al. 2012; Guerrero et al. 2013b; Raftari and Yildirim 2011; Torabi and Yaghoobi 2011)and Laplace-Padé (Bahuguna et al. 2009; Ebaid 2011; Gkdoan et al. 2012; Jiao et al. 2002;Khan and Faraz 2011; Merdan et al. 2011; Momani et al. 2009; Momani and Ertrk 2008;Sweilam and Khader 2009; Tsai and Chen 2010) resummation methods are used in literatureto enlarge the domain of convergence of solutions.

On one side, the Padé resummation method consists of applying the Padé approximant oforder [N/M] to the power series solution. N and M are arbitrarily chosen, but they shouldbe of smaller value than the order of the power series. On the other side, the Laplace-Padéresummation method for power series solutions can be recast as follows:

(1) First, Laplace transformation is applied to power series (2).(2) Next, s is substituted by 1/t in the resulting equation.(3) After that, we convert the transformed series into a meromorphic function by forming its

Padé approximant of order [N/M]. N and M are arbitrarily chosen, but they should be ofsmaller value than the order of the power series. In this step, the Padé approximant extendsthe domain of the truncated series solution to obtain better accuracy and convergence.

(4) Then, t is substituted by 1/s.(5) Finally, using the inverse Laplace s transformation, we obtain the modified approximated

solution.

This process is known as the Laplace-Padé series method (LPSM).

4 The model of the evolution of smoking habit in Spain

This model was presented in Guerrero et al. (2011, 2013b) to describe and predict the evolutionof the smoking habit in Spain and to quantify the impact of the Spanish smoke-free law of2006.

The following system of ordinary differential equations models the dynamics between thedifferent subpopulations considered.

n − μ(1 − n) + βn(s + c) = 0,

s − βn(s + c) − ρe − αc + (γ + λ + μ)s = 0,

c − γ s + (α + δ + μ)c = 0,

e − λs − δc + (ρ + μ)e = 0, (3)

where the dots denote differentiation with respect to t .The subpopulations included in the model are: n is the proportion of the total population

who has never smoked, s is the proportion of people who smoke less than 20 cigarettes perday, c is the proportion of individuals who smoke more than 20 cigarettes per day and e isthe proportion of ex-smokers.

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The parameter μ denotes birth rate in Spain; β denotes the transmission rate due to thesocial pressure to adopt smoking habit; ρ expresses the rate at which ex-smokers return tosmoking; α is the rate at which an excessive smoker becomes a normal smoker by decreasingthe number of cigarettes per day; γ is the rate at which normal smokers become excessivesmokers by increasing the number of cigarettes per day; λ denotes the rate at which normalsmokers stop smoking and δ is the rate at which excessive smokers stop smoking. Thepopulation is constant and it has been normalized to unity, then:

n + s + c + e = 1, (4)

for any instant of time.The asymptotic behaviour of (3) is obtained considering n = 0, s = 0, c = 0, and

e = 0, and solving the obtained system of equations, resulting in two equilibrium points:smoking-free equilibrium (SFE) and smoking endemic equilibrium (SEE), which are SFE =(1, 0, 0, 0) and SEE = (n∗

SEE, s∗SEE, c∗

SEE, e∗SEE) where

n∗SEE = (μ2 + (λ + α + δ + ρ + γ )μ + (α + δ + γ )ρ + (γ + λ)δ + λα)μ

β(α + δ + μ + γ )(ρ + μ),

s∗SEE = η2(α + δ + μ)

η1,

c∗SEE = η2γ

η1,

e∗SEE = η2(λμ + (γ + λ)δ + λα)

η1(ρ + μ), (5)

with

η1 = [μ2 + (λ + α + δ + ρ + γ )μ

+(λ + ρ + γ )δ + ργ + α(λ + ρ)](α + δ + μ + γ )β,

η2 = −[μ3 + (λ + ρ − β + α + δ + γ )μ2 + ((λ − β + γ + ρ)δ + (ρ − β)γ

+(λ − β + ρ)α − βρ)μ − βρ(α + δ + γ )]. (6)

Studying the eigenvalues of the Jacobian matrix associated with the system (3) we cananalyze which one of the two equilibrium points is asymptotically stable and which one isunstable. Moreover, since we are going to use the values of the parameters given in Guerreroet al. (2013b), we can conclude that the point SEE is the stable one. Therefore, our solutionhas to tend asymptotically to the SEE point. For more details about this question, see Guerreroet al. (2013a).

5 Approximated series solution

According to the series method, we propose the following solution of order m

n(t) =m∑

k=0

nkt i , s(t) =m∑

i=0

si ti ,

c(t) =m∑

k=0

ci ti , e(t) =

m∑

i=0

ei ti , (7)

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Approximated solutions for model of smoking habit in Spain

where the initial approximations are

n0 = n(0),

s0 = s(0),

c0 = c(0),

e0 = e(0). (8)

Substituting (7) into (3), we obtain

m−1∑

k=1

knktk−1 = μ

(1 −

m∑

k=0

nktk

)− β

(m∑

k=0

nktk

) (m∑

k=0

(sk + ck)tk

),

m−1∑

k=1

ksktk−1 = β

(m∑

k=0

nktk

) (m∑

k=0

(sk + ck)tk

)+ ρ

m∑

k=0

ektk

+ α

m∑

k=0

cktk − (γ + λ + μ)

m∑

k=0

sk tk,

m−1∑

k=1

kcktk−1 = γ

m∑

k=0

sk tk − (γ + δ + μ)

m∑

k=0

cktk,

m−1∑

k=1

kektk−1 = λ

m∑

k=0

sk tk + δ

m∑

k=0

cktk − (ρ + μ)

m∑

k=0

ektk .

(9)

Next, we equate the terms of the same order, resulting in

n1 = μ(1 − n0) − βn0(s0 + c0),

s1 = βn0(s0 + c0) + ρe0 + αc0 − (γ + λ + μ)s0,

c1 = γ s0 − (γ + δ + μ)c0,

e1 = λs0 + δc0 − (ρ + μ)e0, (10)

and

2n2 = −μn1 − β[n0(s0 + c0) + n1(s0 + c0)],2s2 = β[n0(s0 + c0) + n1(s0 + c0)] + ρe1 + αc1 − (γ + λ + μ)s1,

2c2 = γ s1 − (γ + δ + μ)c1,

2e2 = λs1 + δc1 − (ρ + μ)e1, (11)

for powers t0 and t1, respectively.For the rest of coefficients, we find the following recursive formula

nk+1 = 1

1 + k

⎡

⎣−μnk − β

m∑

p=k

n p(sk−p + ck−p)

⎤

⎦ ,

sk+1 = 1

1 + k

⎡

⎣β

m∑

p=k

n p(sk−p + ck−p) + ρek + αck − (γ + λ + μ)sk

⎤

⎦,

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ck+1 = 1

1 + k[γ sk − (γ + δ + μ)ck],

ek+1 = 1

1 + k[λsk + δck − (ρ + μ)ek], (12)

for k ≥ 1.Therefore, (12) allows us to obtain the series solution in powers of t for any order m.

6 Padé and Laplace-Padé resummation methods

To perform the resummation methods, we set the values of the parameters as in Guerrero et al.(2013b): μ = 0.01 years−1, ρ = 0.0425 years−1, β = 0.0381 years−1, β = 0.0381 years−1,α = 0.1244 years−1, γ = 0.1175 years−1, λ = 0.0498 years−1 and δ = 0.0498 years−1.Moreover, the initial conditions are chosen as: n(0) = 0.5045, s(0) = 0.2059, c(0) = 0.1559and e(0) = 0.1337, as reported in Guerrero et al. (2013b). Then, from (12), we obtain thefollowing solution 20th order approximation

n(t) = 0.50450000 − 0.0019993207t + 0.00011026408t2 − 0.0000036817027t3

+0.000000096547532t4 − 0.0000000022242078t5 + 4.8437208 × 10−11t6

−1.0350119 × 10−12t7 + 2.1771775 × 10−14t8 − 4.4505566 × 10−16t9

+8.7339077 × 10−18t10 − 1.6342918 × 10−19t11 + 2.9017298 × 10−21t12

−4.8437592 × 10−23t13 + 7.4350914 × 10−25t14 − 9.9258933 × 10−27t15

+9.5586875 × 10−29t16 + 1.2390588 × 10−31t17 − 4.0553722 × 10−32t18

+1.4784579 × 10−33t19 − 4.0334550 × 10−35t20,

s(t) = 0.20590000 − 0.0044755390t + 0.00024884128t2 − 0.000012284248t3

+0.00000063314928t4 − 0.000000032767620t5 + 0.0000000015631567t6

−6.6227760 × 10−11t7 + 2.4834689 × 10−12t8 − 8.3079136 × 10−14t9

+2.5041966 × 10−15t10 − 6.8647266 × 10−17t11 + 1.7252816 × 10−18t12

−4.0029155 × 10−20t13 + 8.6247171 × 10−22t14 − 1.7345687 × 10−23t15

+3.2708638 × 10−25t16 − 5.8059941 × 10−27t17 + 9.7357472 × 10−29t18

−1.5471402 × 10−30t19 + 2.3368646 × 10−32t20,

c(t) = 0.15590000 − 0.0045235300t + 0.00015367920t2 + 0.00000031038033t3

−0.00000037514280t4 + 0.000000028699268t5 − 0.0000000015227668t6

+6.6309221 × 10−11t7 − 2.4994901 × 10−12t8 + 8.3579298 × 10−14t9

−2.5157106 × 10−15t10 + 6.8876090 × 10−17t11 − 1.7294192 × 10−18t12

+4.0098432 × 10−20t13 − 8.6353979 × 10−22t14 + 1.7360297 × 10−23t15

−3.2724281 × 10−25t16 + 5.8065162 × 10−27t17 − 9.7320256 × 10−29t18

+1.5455734 × 10−30t19 − 2.3324180 × 10−32t20,

e(t) = 0.13370000 + 0.010998390t − 0.00051278455t2 + 0.000015655570t3

−0.00000035455400t4 + 0.0000000062925616t5 − 8.8827250 × 10−11t6

+9.5354971 × 10−13t7 − 5.7505700 × 10−15t8 − 5.5106120 × 10−17t9

+2.7801100 × 10−18t10 − 6.5396364 × 10−20t11 + 1.2357333 × 10−21t12

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Approximated solutions for model of smoking habit in Spain

−2.0840615 × 10−23t13 + 3.2457857 × 10−25t14 − 4.6820917 × 10−27t15

+6.0836875 × 10−29t16 − 6.4611765 × 10−31t17 + 3.3288889 × 10−33t18

+8.8347368 × 10−35t19 − 4.1332500 × 10−36t20. (13)

To enlarge the domain of convergence, we apply a Padé approximant (Baker 1975) to (13)to obtain

n(t)[8/8] = (0.50449998 + 0.017274274t − 0.00030937773t2

−0.000033327548t3 − 0.00000097656272t4 − 0.000000016082178t5

−0.00000000016936989t6 − 1.1596923 × 10−12t7

−4.3920197 × 10−15t8)/1,

1 = 0.99999997 + 0.038203359t − 0.00068039852t2 − 0.000069808996t3

−0.0000019762218t4 − 0.000000032319349t5 − 0.00000000033869281t6

−2.2556172 × 10−12t7 − 8.5053029 × 10−15t8,

s(t)[10/10] = (0.20590000 − 0.12171188t − 0.0074901950t2

−0.000036696032t3 + 0.000021046910t4 + 0.0000011319850t5

+0.000000039940480t6 + 0.00000000071857987t7

+9.8768910 × 10−12t8

+4.0691945 × 10−14t9 + 1.7800146 × 10−16t10)/2,

2 = 1.0 − 0.56938483t − 0.049962802t2 − 0.00051644392t3

+0.00011433093t4 + 0.0000075362210t5 + 0.00000024423564t6

+0.0000000047931141t7 + 5.6947603 × 10−11t8 + 3.5636890 × 10−13t9

+6.8478700 × 10−16t10,

c(t)[7/7] = (0.15590000 + 0.015745924t + 0.00076149012t2

+0.000027274768t3 + 0.00000056591323t4 + 0.0000000097728885t5

+5.8352741 × 10−11t6 + 3.5093886 × 10−13t7)/3,

3 = 0.99999999 + 0.13001574t + 0.0076712060t2 + 0.00026738041t3

+0.0000059737020t4 + 0.000000085942960t5 + 0.00000000073959329t6

+2.9314471 × 10−12t7,

e(t)[7/7] = (0.13370000 + 0.016494265t + 0.0000086701166t2

−0.00000021344924t3 − 0.000000044755062t4

−0.00000000087169957t5 − 4.1776825 × 10−12t6

−2.4269335 × 10−14t7)/4,

4 = 1.0 + 0.041106021t + 0.00051873320t2 − 0.0000037076810t3

−0.00000020166319t4 − 0.0000000029490808t5

−2.2606150 × 10−11t6 − 8.7346224 × 10−14t7, (14)

where different orders of Padé approximants are required to obtain the best accuracy for eachone of the variables.

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Now, we apply the Laplace-Padé resummation method to n(t) from (13). First, Laplacetransformation is applied to (13) and then 1/t is written in place of s in the equation. After-wards, Padé approximant ([6/5] for n(t) and s(t), [5/4] for c(t) and e(t)) is applied and 1/sis written in place of t . Finally, using the inverse Laplace s transformation, we obtain themodified approximated solution

n(t)[6/5] = 0.51086151

−0.59911800 × 10−4 exp(−0.21704764t) cos(0.034646335t)

+0.83968340 × 10−4 exp(−0.21704764t) sin(0.034646335t)

+0.47169326 × 10−3 exp(−0.20129121t)

+0.024786891 exp(−0.091934389t)

−0.031560192 exp(−0.011358812t), (15)

Next, we apply the same procedure for the rest of the variables of (13), resulting in

s(t)[6/5] = 0.15307771 + 0.001406921 exp(−0.30192977t)

+0.57701934 × 10−4 exp(−0.28788644t)

+0.037344784 exp(−0.09336449t)

+0.54005316 × 10−2 exp(−0.083974802t)

+0.86123492 × 10−2 exp(−0.010908388t),

c(t)[5/4] = 0.097900879 − 0.0014605035 exp(−0.30159029t)

−0.68930622 × 10−3 exp(−0.18863972t)

+0.054460116 exp(−0.092296567t)

+0.56888175 × 10−2 exp(−0.011874625t),

e(t)[5/4] = 0.23775395 + 1.8641738 × 10−8 exp(−0.58305125t)

+0.21834747 × 10−3 exp(−0.19187867t)

−0.12182142 exp(−0.092205964t)

+0.017549108 exp(−0.010961758t). (16)

7 Discussion

Figures 1, 2, 3 and 4 show a comparison between the Fehlberg fourth–fifth order Runge–Kutta method with degree four interpolant (RKF45) (Enright et al. 1986; Fehlberg 1970)solution (built-in function of Maple software) for the dynamics of smoking habit in Spain(3) and the analytic approximations (13), (14), (15) and (16). Moreover, in order to obtain agood numerical reference, the accuracy of RKF45 was set to an absolute error of 10−7 andrelative error of 10−6.

The SM solution (13) is easily obtained by a straightforward procedure. However, asdepicted in Figs. 1, 2, 3 and 4 and Table 1, the power series solution diverges for largeperiods of time. Therefore, Padé resummation method was successfully applied to powerseries (13), resulting in a good agreement with RKF45 results for a period of 200 years.

To enlarge the convergence even more, Laplace-Padé resummation method was appliedto (13), resulting in a good accuracy of the approximation for a period longer than 200 years.Moreover, from Table 1, we can observe a comparison of the exact asymptote (5) of model

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Approximated solutions for model of smoking habit in Spain

Fig. 1 Solution of series method with 20 terms for n(t) (diamonds), RKF45 solution (diagonal cross), SM-Padé solution (solid line) and LPSM solution (dots)

Fig. 2 Solution of SM method with 20 terms for s(t) (diamonds), RKF45 solution (diagonal cross), SM-Padésolution (solid line) and LPSM solution (dots)

(3) and the proposed approximations evaluated at t = ∞, resulting in LPSM solution beingthe most accurate approximation in the range of t = [0,∞] years. To illustrate this, we plotin Fig. 5 a comparison between the LPSM solution, the HAM-Padé result (Guerrero et al.2013b) and the exact solution for s(t).

Finally, as aforementioned, the resummation methods are a powerful tool to enlarge thedomain of convergence of power series solutions that can accurately describe asymptoticnon-linear problems like the evolution of the prevalence of the smoking habit in Spain.

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H. Vazquez-Leal, F. Guerrero

Fig. 3 Solution of series method with 20 terms for c(t) (diamonds), RKF45 solution (diagonal cross),SM-Padé solution (solid line) and LPSM solution (dots)

Fig. 4 Solution of series method with 20 terms for e(t) (diamonds), RKF45 solution (diagonal cross),SM-Padé solution (solid line) and LPSM solution (dots)

Table 1 Exact and approximations of asymptote for model of smoking habit in Spain

Variable Exact (5) SM (13) SM-Padé (14) LPSM(15), (16)

HAM(Guerrero et al.2013b)

HAM-Padé(Guerrero et al.2013b)

n(∞) 0.51143607 −∞ 0.50450001 0.51085122 −∞ 0.51073075

s(∞) 0.15308033 −∞ 0.20590000 0.15322657 +∞ 0.57420183

c(∞) 0.097648961 +∞ 0.15590000 0.097900879 −∞ 0.08620481

e(∞) 0.23783464 −∞ 0.13370000 0.23775484 −∞ 0.24404096

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Approximated solutions for model of smoking habit in Spain

Fig. 5 Comparison between LPSM solution (circles), HAM-Padé solution (dots) and RKF45 solution (solidline) for s(t)

8 Concluding remarks

In this paper, powerful analytical methods, SM, SM-Padé and LPSM methods, are applied toconstruct approximated analytical solutions for the model of the evolution of smoking habitin Spain. The series method (SM) provides solutions in the form of fast convergent series witheasily computable components. After that, we applied Padé and Laplace-Padé resummationmethods to successfully enlarge domain of convergence. The numerical experiments anderror analysis are presented to support the theoretical results. Our solutions agree well withthe pure numerical solutions and are better than the approximated solution obtained usingHAM-Padé approximant in Guerrero et al. (2013b). Finally, the comparison tables also showthat LPSM provides the most accurate solution for a period of t = [0,∞] years.

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