Research Journal of Recent Sciences _________________________________________________ ISSN 2277-2502 Vol. 2(1), 91-96, January (2013) Res.J.Recent Sci. International Science Congress Association 91 Application of Homotopy Analysis Method to SIR Epidemic ModelVahdati S.1 , Tavassoli Kajani M.2 and Ghasemi M.Department of Mathematics, Khansar Faculty of Mathematics and Computer Science, University of Isfahan, Isfahan, IRAN Department of Mathematics, Khorasgan Branch, Islamic Azad University, Isfahan, IRAN Department of Applied Mathematics, Faculty of Mathematical Sciences,Shahrekord University, Shahrekord, IRAN Available online at: www.isca.in Received 12th October 2012, revised 9th November 2012, accepted 30th November 2012Abstract In this article, the problem of the spread of a non-fatal disease in a population which is assumed to have constant size over the period of the epidemic is considered. Mathematical modeling of the problem leads to a system of nonlinear ordinary differential equations. Homotopy analysis method is employed to solve this system of nonlinear ordinary differential equations. Keywords: Homotopy Analysis method, System of non-linear differential equations; PACS: 02.60.Lj, 02.60.Cb, 87.10.Ed IntroductionIn 1992, Liao1,2 proposed a new analytical technique; namely the Homotopy Analysis Method (HAM) based on homotopy of topology. However, in Liao's PhD dissertation, he did not introduce the auxiliary parameter  , but simply followed the traditional concept of homotopy to construct the following one-parameter family of equations 0,(1 + - (1) where L is an auxiliary linear operator, N is a nonlinear operator related to the original nonlinear problem and p is the embedding parameter. An improved two parameters family of equations was proposed to avoid divergence of solution by introducing an auxiliary parameter  3, 4. ).(1 (2) where is an initial guess. Using the definition of Taylor series with respect to the embedding parameter p (which is a power series of p ), Liao gave a general equations for high-order approximations. He5, 6 constructed the one-parameter family of equations 0,(1 + - (3) which is exactly the same as Liao's early one-parameter family equation (1), and is a special case of Liao's modified two-parameter equation (2) when 1 = -  . The problem of spreading of a non-fatal disease in a population which is assumed to have constant size over the period of the epidemic is considered in. At time t suppose the population consist of )( susceptible: those so far uninfected and therefore liable to infection; infective: those who have the disease and are still at large; )( who are isolated, or who have recovered and are therefore immune. Assume there is a steady constant rate between susceptible and infective and that a constant proportion of these constant result in transmission. Then in time d , d of the susceptible become infective, where xy d b d - (4) and b is a positive constant. If g is the rate at which current infective become isolated, then xy d g d b d - (5) The number of new isolates d is given by d g d (6) Now let ¥ ® t d . Then the following system determines the progress of the disease: Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(1), 91-96, January (2013) Res. J. Recent Sci. International Science Congress Association 92 dtdzxydtdyxydtdx (7) with initial conditions, (0), (0) (0). Homotopy Analysis Method (HAM)In this paper we apply the homotopy analysis method to the problem. Consider the following equation: )]( J where N is a nonlinear operator, t denotes the independent variable, J is an unknown function. The so-called HAM's zero-order deformation can be obtained )]])(=)];(()(1[0] (8) where [0,1] Î is the embedding parameter, ¹ is a nonzero auxiliary parameter,)( ¹ is an auxiliary function, L is an auxiliary linear operator, [0] is an initial guess of J , ;( f is an unknown function. When and 1, then ),;1) ),;0)[0]respectively. Thus, as p increases from 0 to 1, the solution ;( f varies from the initial guess [0] to the solution J . Expanding ;( f in Taylor series with respect to p , one has ;((01=[0]+¥ (9) where ;((0pkkkpptkt¶¶fJ (10) If the auxiliary linear operator, the initial guess, the auxiliary parameter  , and the auxiliary function are properly chosen, the series (9) converges at , thus ),,01=[0]+¥ (11) which must be one of the solutions of the original nonlinear equation, as proved by Liao 8. As pointed by Liao, the auxiliary parameter  can be employed to adjust the convergence region of homotopy analysis solution. In general, by means of the so-called  curve, it is straightforward to choose an appropriate range for  which ensures the convergence of the solution series. According to the equation (10), the governing equation can be deduced from the zero-order deformation equation (8). Define the vector { } .)(),(),((0[1][0]  Differentiating equation (8) times with respect to the embedding parameter p and then setting and finally dividing them by , we have the so-called th-order deformation equation ),)()]()((11]  (12) where )];((1)! (13) and 1.1,1,0, (14) It should be emphasized that 0tkJ for ³ is governed by the linear equation (10) with the linear boundary conditions that comes from the original problem, which can be solved easily by symbolic computation software such as Matlab, Maple or Mathematica. In this paper all calculations were accomplished using Maple software where the long format and the double precision have been used for high accuracy results. Solution of the epidemic model by HAMBy applying inverse of the operator dt (.), which is integration operator dt(.) to each equations in the system (7) we derive Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(1), 91-96, January (2013) Res. J. Recent Sci. International Science Congress Association 93 (0))]](0)(0)dtdtdt (15) Let ( ) ( ) and ( ) are the initial guesses of ( ) ( ) and ( ) respectively. According to equation (8), the HAM's zeroth-order deformation for equation (15) will be ()()()()()()()()()()()()()()()[]()()()()()()dtdtdt(1(1(1 (16) For and , we can write ( ) ( ) ( ) ( ) ()()()()()()()(),1,0,1,0,1,0 (17) Considering Maclaurin series of ( ) ( ) and ( ) corresponding to p , one has () () () +¥,(,(,((01=0][01=0][01=0 (18) which ,()(,()(,()((00=][00=][0pkkkpkkkpkkkpptZtzpptYtypptXtx¶¶¶¶¶¶ (19) Define () () () . ! )(=, ! )(=, ! )(=][0][0][0 k tztz k tyty k (20) If , using equation (18), then ()()()()()()(,(,+¥+¥+¥ (21) Define the vectors ( ) ( ) ( ) ( ) { } ()()()(){}()()()(){}  (22) Thus we obtain the th-order deformation equation ( ) ( ) ()()()())..),,),  (23) from equation (13) and (16), we have ()()()()()()()()()()()()()()()()()()1=1=dtdtdt(24) Considering equation (14), (16), (23) and (24) we can find the recursive expression of epidemic model ()()()()()()()()()()()()()()()()()()()()()()()()1=1=dtdtdt(25) Convergence Theorem: Theorem 3.1 If the solution series in equation (21) converge, where ( ) ( ) and ( ) is governed by equation (23) under definition (14) and (24), they must be the solution of equation (15). Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(1), 91-96, January (2013) Res. J. Recent Sci. International Science Congress Association 94 Proof. If the series ()()()+¥+¥+¥converge, we can write ()()()()()()+¥+¥+¥ (26) and they hold ( ) ( ) ( ) 0.lim0,lim0,lim (27) Using the definition of we have ()()()()()()()()()()()()()()()()()()),==][,==][12312111=12312111=12312111=tzzzzzzzztztztyyyyyyyytytytxxxxxxxxtxtxnnnkkknknnnkkknknnnkkknkccc (28) which gives us, according to (27) ()()()()()()()()()0.limm0,limm0,limm11=11=11=tztztztytytytxtxtxkkkkkkkkkkkkkkkkkk¥®-¥+¥®-¥+¥®-+¥ (29) Furthermore, using the above expression and the definition of L , we have ()()()()()()()()()()()()0..=][0,,=][0,,=][11=11=11=11=11=11=tztztztztytytytytxtxtxtxkkkkkkkkkkkkkkkkkkkkkkkk-¥+-¥+-¥+-¥+-+¥+¥ (30) From the above expression and equation (23), we obtain ()()()()()())0,,0,,11=11=11=11=11=11=-¥+-¥+-¥+-¥+-+¥+¥ (31) which gives, since ¹ , that 0.0,0,+¥+¥+¥ (32) From (24), we have ()()()()()()()()()()()()()()()()()()dtdtdtdt(33) From equations (32) and (33) we have ()()()()dtUsing similar procedure we obtain ()()()()()()()()()dtdtTherefore, according to the above two equations, ( ) ( ) and ( ) must be exact solution of equation (15). Numerical exampleFor numerical results the following values, for parameters, are considered: 20, Initial population of )(, who are susceptible, 15, Initial population of , who are Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(1), 91-96, January (2013) Res. J. Recent Sci. International Science Congress Association 95 infective, 10, Initial population of )(, who are immune, 0.01 = b , Rate of change of susceptible to infective population, 0.02 g , Rate of change of infective to immune population. Figures 1-3 show the  -curves obtained from the 12th-order, 8th-order and 6th-order HAM approximation solutions of equation (21). From these figures, the valid regions of  correspond to the line segments nearly parallel to the horizontal axis. Figure-1 The  -curve of (1) given by (21) Dotted line: 12th-order approximation; Dashed line: 8th-order approximation; Solid line: 6th-order Figure-2 The  -curve of (1) given by (21) Dotted line: 12th-order approximation; Dashed line: 8th-order approximation; Solid line: 6th-order Figure-3 The  -curve of (1) given by (21) Dotted line: 12th-order approximation; Dashed line: 8th-order approximation; Solid line: 6th-order These results according to some values of  (the abscissa of intersection points in figures 1-3) are plotted in figures 4-6. As the plots show while the number of susceptible increases the population of who are infective decreases in the period of the epidemic, meanwhile the number of immune population increases. Figure-4 Plot of twelve terms approximation for )( when 1.0053219 - Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(1), 91-96, January (2013) Res. J. Recent Sci. International Science Congress Association 96 Figure-5 Plot of twelve terms approximation for when 1.0047109 - Figure-6 Plot of twelve terms approximation for )( when 1.0639236 - ConclusionHomotopy analysis method is known to be a powerful device for solving many functional equations such as ordinary, partial differential equations, integral equations and so many other equations. In this article, we used homotopy analysis method for solving a system of differential equation which describe SIR model for an epidemic disease. Using the concept of  -curve, we found the convegence aera for the series solution of the problem. Numerical examples also provided to show the ability and efficiency of the method. References1.Liao S.J., The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, (1992)2.Liao S.J., A kind of linear invariance under homotopy and some simple applications of it in mechanics, Bericht Nr. 520. Institute fuer Sciffbau der Universitaet Hamburg, (1992)3.Liao S.J., Numerically solving nonlinear problems by the homotopy analysis method, Comput. Mech., 20, 530-540 (1997)4.Liao S.J., A kind of approximate solution technique which does not depend upon small parameters (II): an application in fluid mechanics, Int. J. Non-linear Mech., 32, 815-822 (1997)5.He J.H., Homotopy perturbation technique, Comput Methods Appl. Mech. Engng., 178, 257-262 (1999)6.He J.H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. Nonlinear Mech., 35(1), 37-43 (2000)7.Jordan D.W. and Smith P., Nonlinear Ordinary Differential Equations. third ed, Oxford University Press, (1999)8.Liao S.J., Homotopy Analysis Method: A New Analytical Technique for Nonlinear Problems, Commun. Nonlinear Sci. Numer. Simul., 2(2) 95-100 (1997)9.Liao S.J., Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman and Hall/CRC Press, London/Boca Raton (2003)