Research Journal of Recent Sciences ________________________________________________ ISSN 2277-2502Vol. 2(8), 66-71, August (2013) Res.J.Recent Sci. International Science Congress Association 66 Analytical Solution of the Leptospirosis Epidemic model by Homotopy Perturbation methodMuhammad Altaf Khan, Islam S., Murad Ullah, Sher Afzal KhanZaman G.4 and Syed Farasat SaddiqDepartment of Mathematics, Abdul Wali Khan University Mardan, PAKISTAN Department of Mathematics, Islamia College University, Peshawar, PAKISTAN Department of Computer Science, Abdul Wali Khan University Mardan, PAKISTAN Department of Mathematics, University of Malakand Chakdara, Dir (lower), PAKISTAN Available online at: www.isca.in Received 24th February 2013, revised 25th March 2013, accepted 21st April 2013 AbstractIn this paper, we consider a mathematical model of leptospirosis disease consisting of differential equations. We apply the Homotopy perturbation method to the proposed model to find both the analytic and approximate solutions. From our solutions, we obtained that Homotopy perturbation method is one of the most important method, like just few perturbation terms are sufficient for obtaining a reasonable accurate solution. The solution obtain from this method is good as compared to other standard numerical methods. Keywords: Leptospirosis, mathematical model, Homotopy Perturbation Method, numerical simulations Introduction The infection of Leptospirosis disease is a globally important. Because the infection of the disease occurs in developed and industrialized countries as well as rural regions in the world. The people infected from this disease easly are rice planters, sewer cleaners, workers cleaning canals, agriculture labors. In order to understand the dynamics of this infectious disease, several authors proposed different mathematical models1-4. Pongsuumpun et al., developed a mathematical model to study the dynamical behavior of Leptospirosis disease. Triampo et al., considered a deterministic models for the transmission of Leptospirosis disease by collecting some real data. Zaman, studied the dynamical behavior of Leptospirosis and applied optimal control theory to control the spread of this disease in the community. Most of the biological problems in the form of mathematical models are inherently nonlinear. Therefore it’s not only difficult but always impossible to find the exact solutions that represent the complete biological phenomena. So, the scientists are in search to find such numerical methods or perturbations method to find the exact solution and approximate solution to these non-linear problems. In the numerical methods, stability and convergence should be considered so as to avoid divergence or inappropriate results. While, in the analytical perturbation method, we need to exert the small parameter in the equation. Therefore, finding the small parameter and exerting it into the equation are difficulties of this method. However, there are some limitations with the common perturbation method, like the common perturbation method is based upon the existence of a small parameter, which is difficult to apply to real world problems. Therefore, many different powerful mathematical methods have been recently introduced to vanish the small parameters, such as artificial parameter method8,9. The Homotopy Analysis Method (HAM) is one of the wellknown methods to solve the nonlinear equations. In the last decade, the idea of homotopy was combined with perturbation. The fundamental work was done by Liao and He. This method involves a free parameter, whose suitable choice results into fast convergence. First time He10, introduced Homotopy Perturbation Method (HPM) and its application in several problems11,12. Ali et al13, presented the solution of multi points boundary values by using Optimal Homotopy Analysis Method (OHAM). These methods are independent of the assumption of small parameter as well as they covered all the advantages of the perturbation method. Some other relevant work can be find by other researchers14-17. The motivation of this paper is to present the application of the analytic homotopy perturbation method (HPM) to solve a model of epidemic leptoapirosis disease1. First, we formulate our problem and then apply the HPM to find the analytical as well as numerical solutions. Finally, we also estimates the parameter in the model for the numerical simulation. The paper is organized as follows. Section 2 is devoted to the basic idea of HPM and the mathematical formulation of the model. In Section 3 the model is solve by HPM. We present the numerical solution and discussion in section 4. Basic idea of Homotopy Perturbation Method (HPM) and model frame work: In this section, first we explain the Homotopy perturbation method in detail and then we apply the technique of HPM to our proposed Leptospirosis epidemic model. HPM was first time introduced by He8,9 , for solving the non linear differential equations.  \n (1) Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(8), 66-71, August (2013) Res. J. Recent Sci. International Science Congress Association 67 subject to the boundary conditions \r\r  \n (2) Here B represents the general differential operator, is the boundary operator, is the analytic function, is the boundary of the domain , \r represents the differentiation along the normal vector L drawn outward. The operator B is divided in two parts, H is linear and K is nonlinear to get the following equation  (3) Define the homotopy , that satisfies     (4) Which can be written as  (5) Where mo shows the initial approximation of (5) and p is the embedding parameter such that\n. It is obvious that  For p=0 we get,  while for p=1 we obatin  . Applying the perturbation technique by considering parameter p for small value then the solution of equation (4) can be obtained in p series is given by % (6) when p approaches 1 the equation (4) becomes the original equation (3) and (7) becomes the approximate solution of (3) is given by &'("% (7) The series (7) is convergent for most of cases see [8,9]. In order to formulate our problem, we assume that S(t) represents number of susceptible human at time t; I(t) represents number of infected human in the population, which is infected from the leptospirosis disease at time t; R(t) represents number of human in the population which is recovered at time t. The total population size is N(t)=S(t)+I(t)+R(t). For vector population, let P(t) are susceptible vector and M(t) are infected vector at time t. The total population size of vector population is denoted by N(t) with N(t)=P(t)+M(t). By the interaction of both human and vector population we get the following system of five differential equations is given by: )+ , - . .21+ ..456769 52 (8) :+ , 5 ./+ . 5 4With the initial conditions ;1;3; :; /; represents the growth rate of human population. The direct transmission between susceptible human and infected vector is represented by . The transmission rate of the vector is shown by . The natural death rate for the human is . The disease death rate for the human is represented by . The natural mortality rate for vector population is represents the disease related death rate for vector. # represent the population growth rate for the vector. $ is the transmission coefficient between susceptible vector and infected human. The rate of recovery from the infection is shown by . The individuals are susceptible again at . Now, we apply the homotopy perturbation techniques to our model (8). We assume for simplifications )1133:: �? //Now we define the operator @69 A The initial data we consider is given by @) @)B, - .)/ .)123 @)C@1 @1+.)/.)1 -451 @1+ ) , @3 @3+51 23 @3+ (11) @: @:+, 5 .:1 @:+@/ @/+.:1 5/ 4/ @/)1133:: �?// (12) Assume the solution of the system (11) in the form))E11E33E (13) ::E//EBy considering equation (13) in equation (11), comparing the same coefficient, we obtain @) @)+@1 @1+@3 @3+ (14) @: @:+@/ @/+AAnd Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(8), 66-71, August (2013) Res. J. Recent Sci. International Science Congress Association 68 @), . . -2 @)@1.. -451 @1@35 -23 @3 (15) @:, 5 . @:@/.4 @/With the initial conditions 13:�?/ (16) And @) .B))C .)) -2@).B))C.)) -451@35 -23 (17) @: 5 .::@/.:: 4With the initial conditions 13:�?/ (18) In similarway, we get @) .B)))C ))) -2@1.B))) (19) .))) -451@35 -23@: 5 .:::@/.:::4with initial condition 13:�? / (20) To find the solution, we consider p=1 inthesystem (11), we get )))E111E333E (21) :::E///EIn order to we obtain the solution to the zero order problem, we consider the following cases. Zeroth order Problem or P0 F1G3HH�? /H First order Problem or P1 - . .2+. -45+ -2+ (22) 5 .+4+Numerical Results In this section, we discuss the numerical solution of the proposed model. First, we solve the model numerically and then discuss these results. For numerical simulation we consider the parameter values presented in table 1. The numerical results are presented in figure-1, 2 and 3 show the population of susceptible human, infected human and recovered human, respectively. Figure-4 and 5 show the population of susceptible vector and infected vector. Second order solution or P2 F" 1G# 3$ KHHJ �? LHI  (23)  -M - . .2N .O4  - . .2 .O.4  - . .2 P2M2N . O4  - . .2 P .O.4  - . .2 45M. -45N  Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(8), 66-71, August (2013) Res. J. Recent Sci. International Science Congress Association 69 5. -45 T 22 T: 5 5 . T .R 5 . . -45 T/.R 5 . -45 4M4N A (24) Figure-1 The plot represents the population of susceptible human in the model Figure-2 The represents the population of infected human in the model Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(8), 66-71, August (2013) Res. J. Recent Sci. International Science Congress Association 70 Figure-3 The represents the population of recovered human in the model Figure-4 The plot shows the population of susceptible vector in the model Figure-5 The plot shows population of infected vector in the model Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(8), 66-71, August (2013) Res. J. Recent Sci. International Science Congress Association 71 Table-1 Parameter values used for the Numerical Simulation 4 0 Disease death rate of Vector 0.0094 4 * Disease death rate of a human 0.0000001 , " Recruitment rate of human population 1.6 . " Direct transmission between susceptible human and infected human 0.00005 . $ Transmission between susceptible vector and infected human 0.0078 . # Transmission between susceptible human and infected vector 0.0098 - * A natural death rate of a human 0.0034 , # Birth rate for vector population 1.2 5 0 Natural death rate of vector 0.0017 2 * The rate at which the individuals become susceptible again 0.00067 5 * A recovery rate of infection of human 0.007 Conclusion In this paper, we considered an epidemic model represented the interaction of the leptospirosis infected vector and human population. Leptospirosis is a zoonotic disease which is found mostly areas. The model is formulated and applied the homotopy perturbation technique and the numerical as well as their analytical solution was obtained. The model is solved up to second order by the Homotopy perturbation method. The homotopy perturbation method gives a good result for the non-linear system, with a few iterations. 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