Differential Transform and Butcher s fifth order Runge-Kutta Methods for solving the Aedes-Aegypti model

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1 Differential Transform and Butcher s fifth order Runge-Kutta Methods for solving the Aedes-Aegypti model R. Lavanya 1 and U. K. Shyni Department of Mathematics, Coimbatore Institute of Technology, Coimbatore , Tamil Nadu, India. Abstract: This article presents the numerical study of the Aedes-Aegypti predator prey model in population dynamics by the Differential Transform Method (DTM) and the Adomian Decomposition Method(ADM). The significance of these two methods lies in the fact that the solution of the equations can be written in a series form with easily computable terms. Numerical examples are presented and the results are compared with the Butcher s fifth-order Runge- Kutta solutions to illustrate the strength and effectiveness of these methods. AMS subject classification: 34A34, 65L05, 92B05. Keywords: Butcher s method, Differential Transform Method, Adomian Decomposition Method, Predator-prey model, Numerical Analysis. INTRODUCTION The role of arthropods like mosquitoes as vectors responsible for transmission of epidemics remains of great importance worldwide as it causes significant mortality and morbidity. Many of such medically important transmissible diseases of the world share one important characteristic; their transmission from human to human requires a blood feeding arthropod vector that may serve as an intermediate host to the infective organism. One such epidemic, dengue haemorrhagic fever or the Dengue fever ranks as the most important mosquito borne viral disease in the world. Mosquitoes belonging to the genus aedes (Aedes aegypti, Aedes albopictus, and Aedes polynesiensis) play an important part in transmission of dengue. Among them, the primary and the most efficient vector of dengue virus is the species Aedes aegypti which dwell in tropical and subtropical regions all over the world [14]. In addition to the global distribution of mosquitoes, rapid human population growth and increased urbanisation has led to substandard housing, inadequate water supply and waste management systems, and consequently an abundance of mosquito breeding sites. Storage of drinking water and other urban water, containers including plant-pot bases, guttering, tarpaulins, tyres and discarded containers can all collect rainwater and provide habitats for Aedes aegypti larvae. J. H. Arias et al., [13] proposed a predator-prey model to analyse the population dynamics of the Aedes aegypti mosquito which was responsible for the transimition of dengue in Cali, Colombia. Here a predator is voluntarily introduced into the prey breeding habitat where the Aedes aegypti immature stages are already present and constitute the prey population. The possible predators of these mosquitoes include cyclopoid copepods (like Macrocyclops albidus, Mesocyclops aspericornis, Mesocyclops longicetus) [17] and Larvivorous fish (like Gambusia affinis and Poecilia reticulata). Aedes aegypti females lay their eggs on sidewalls of stagnant waterbodies. The eggs furthur hatch into larvae that are transformed into pupae which eventually becomes adult mosquitoes. Only two mosquito stages namely, immature or acquatic (comprising egg, larvae and pupae) and mature or aerial (corresponding of adult female mosquitoes) are considered in the mathematical modelling of our model [18]. 1 Corresponding author lavanya.r@cit.edu.in

2 The Aedes aegypti predator-prey model is represented as follows: ) di = ηam (1 IαI (ω + γ I + ζp) I (1) dt dm = bωi γ M M (2) dt ) dp = (τ + µi) P (1 PαP (3) dt with initial conditions: I(0) = I 0 ; M(0) = M 0 ; P (0) = P 0, (4) where I denotes the number of immature stages at time t, M denotes the number of adult female mosquitoes at time t and P denotes the number of natural predators of immature stages at time t. The parameters of the model are defined as in [13]. The principal interest of this article is to conduct the numerical analysis of Aedes Aegypti system (1) (4) applying two semi-analytical methods, namely Differential Transform Method (DTM) and Adomian Decomposition Method (ADM). Numerical solutions of the same is then computed using the Butcher s fifth order Runge Kutta method [11] and the results are then compared with that of DTM and ADM. This article is organized as follows: The algorithm for the system (1) (4) by Differential Transform Method is presented in Section 1. The decomposition of the system (1) (4) by Adomian Method is presented in Section 2. Numerical examples and its comparitive study is presented in Section 3 and the concluding remarks are presented in Section 4. DIFFERENTIAL TRANSFORM METHOD The Differential Transform Method(DTM) was proposed by Georgii Evg. Pukhov [8] as Taylor Transform in the year 1978 which was furthur developed by him in the name of Differential Transform from , where its mathematical formulae and rules allowed the creation of both analytical and numerical methods for solving linear and non-linear systems. It was then used by by Zhou [19] in the year 1986 to solve both linear and nonlinear initial value problems in electric circuit analysis. The DTM gives exact values of the nth derivative of an analytic function at a point in terms of known and unknown boundary conditions in a fast manner. In short, DTM is a numerical as well as analytical method for solving integral equations, ordinary, partial differential equations and differential equation systems [[2], [9], [10], [15], [16]]. To illustrate the basic idea of this method we define the differential transform of a function f (t) as F (k) = D k {f (t)} = 1 [ d k ] k! dt f (t) k t=t 0 (5) The inverse differential transform of F (k) is defined as: f (t) = F (k)(t t 0 ) k (6) k=0 The above combination gives the Taylor series of f (t). Now applying the Differential Transform Method to our system (1) (4) we have { [ ] I(k + 1) = 1 ηa M(k) 1 k M(l)I(k l) k + 1 α I l=0 (7) } k (ω + γ I ) I(k) ζ P (l)i(k l) M(k + 1) = 1 k + 1 {bωi(k) γ MM(k)} (8) { P (k + 1) = 1 τp(k) τ k P (l)p (k l) (9) k + 1 α P l=0 k + µ I(l)P (k l) µ α P l=0 k l=0 r=0 l=0 } l I(r)P (l r)p (k l) ADOMIAN DECOMPOSITION METHOD The Adomian decomposition method (ADM) developed by G. Adomian [[3], [4], [5], [6]], is a semi-analytical numerical technique for solving ordinary and partial nonlinear differential and integro-differential equations. The important feature of this method is the possibility to separate the linear and nonlinear parts and the use of Adomian polynomials in the solution convergence of the nonlinear portion of the equation instead of linearizing the whole system [[1], [12]]. The solution of the differential equation is then expressed in the form of an infinite series in which each term is determined recurssively [7]

3 Let L be the first order differential operator L = d dt. (10) Assuming L is invertible, L 1 is given by L 1 (.) = t Therefore, by the decomposition method 0 (.)dt. (11) L 1 LI = I(t) I(0) (12) L 1 LM = M(t) M(0) (13) L 1 LP = P (t) P (0). (14) Applying the decomposition method to our system (1) (3) with initial conditions (4) results in the following: [ I(t) = I(0) + ηa L 1 {M} 1 ] L 1 {MI} (15) α I (ω + γ I ) L 1 {I} ζl 1 {PI} M(t) = M(0) + bωl 1 {I} γ M L 1 {M} (16) P (t) = P (0) + τl 1 {P } τ L 1 { P 2} α P (17) + µl 1 {IP} µ α P L 1 { IP 2} Expressing the non-linear terms of the above system as A = MI, B = PI, C = P 2 and D = IP 2,wehave [ I(t) = I(0) + ηa L 1 {M} 1 ] L 1 {A} (18) α I (ω + γ I ) L 1 {I} ζl 1 {B} M(t) = M(0) + bωl 1 {I} γ M L 1 {M} (19) P (t) = P (0) + τl 1 {P } τ L 1 {C} α P (20) + µl 1 {IP} µ α P L 1 {D} Thus the infinite series solution of the unknown functions I(t), M(t) and P (t) are expressed respectively as I(t) = I n (t), M(t) = M n (t) and P (t) = P n (t), while the infinite series of the nonlinear terms are given by A = A n, B = B n, C = C n and D = NUMERICAL EXAMPLES D n. In this section we analyse three examples of our predatorprey system under three different temperature conditions [13]. The numerical solutions will be obtained and the results will be compared graphically. Example 1. When the average temperature is at T AV G = 23.9 C We consider our system (1) (3) with the values of parameters as follows: η = 5.49, a = 0.9, α I = 1, ω = 0.08, γ I = 0.068, ζ = 0.559, b = 0.5, γ M = 0.035, τ = 1, µ = 0.1 and α P = 10. Case 1 : Absence of predators I(0) = I 0 = 0.2; M(0) = M 0 = 0.3; P (0) = P 0 = 0, (21) Case2:Presence of predators I(0) = I 0 = 0.2; M(0) = M 0 = 0.3; P (0) = P 0 = 3, (22) The solution of our given problem under initial conditions (21) and (22) is given in Table 1, Table 2 and Table 3. The graphical comparison of Example 1 is given in Fig. 1 and Fig. 2. Table 1. Comparison of numerical values of I(t)

4 Table 2. Comparison of numerical values of M(t) Table 3. Comparison of numerical values of P (t) FIGURE 1. Comparison graph for Example 1 : Case 1 Example 2. When the minimum temperature is at T MIN = 19 C We consider our system (1) (3) with the values of parameters as follows: η = 2.15, a = 0.9, α I = 1, ω = 0.068, γ I = 0.029, ζ = 0.178, b = 0.5, γ M = 0.036, τ = 1, µ = 0.1 and α P =

5 FIGURE 2. Comparison graph for Example 1 : Case 2 Table 4. Comparison of numerical values of I(t) Case 1 : Absence of predators I(0) = I 0 = 0.2; M(0) = M 0 = 0.3; P (0) = P 0 = 0, (23) Case2:Presence of predators I(0) = I 0 = 0.2; M(0) = M 0 = 0.3; P (0) = P 0 = 3, (24) The solution of our given problem under initial conditions (23) and (24) is given in Table 4, Table 5 and Table 6. The graphical comparison of Example 2 is given in Fig. 3 and Fig. 4. Example 3. When the maximum temperature is at T MAX = 29.8 C We consider our system (1)-(3) with the values of parameters as follows: η = 8.82, a = 0.9, α I = 1, ω = 0.14, γ I = 0.07, ζ = 1.841, b = 0.5, γ M = 0.03, τ = 1, µ = 0.1 and α P = 10. Case 1 : Absence of predators I(0) = I 0 = 0.2; M(0) = M 0 = 0.3; P (0) = P 0 = 0, (25) 13853

6 Table 5. Comparison of numerical values of M(t) Table 6. Comparison of numerical values of P (t) FIGURE 3. Comparison graph for Example 2 : Case 1 Case2:Presence of predators I(0) = I 0 = 0.2; M(0) = M 0 = 0.3; P (0) = P 0 = 1, (26) The solution of our given problem under initial conditions (25) and (26) is given in Table 7, Table 8 and Table 9. The graphical comparison of Example 3 is given in Fig. 5 and Fig

7 FIGURE 4. Comparison graph for Example 2 : Case 2 Table 7. Comparison of numerical values of I(t) P (0) = 0 P (0) = Table 8. Comparison of numerical values of M(t) P (0) = 0 P (0) =

8 Table 9. Comparison of numerical values of P (t) P (0) = 0 P (0) = FIGURE 5. Comparison graph for Example 3 : Case 1 CONCLUSION In this paper we investigated the numerical solutions of the Aedes Aegypti predator-prey population dynamics model by the Differential Transform andadomian Decomposition Methods. A series of numerical examples and the comparison of their solutions with Butcher s fifth order Runge Kutta method indicate that the presented two methods are easy to use and they reduce significant part of our computational time. From these examples we also observe that as the temperature rises to maximum, the number of terms required in the series expansion of both our methods for the convergence of the numerical solution completely depends on the initial number of predators present. Here we consider our predator as copepods and hence the values of parameters are set according to the nature of the predator. Numerical study of our Aedes Aegypti model having Larvivorous fish like Gambusia affinis and Poecilia reticulata as predators can also be carried out as an extension of this work

9 FIGURE 6. Comparison graph for Example 3 : Case 2 REFERENCES [1] K. Abbaoui and Y. Cherruault (1994) Convergence of Adomian s method applied to differential equations, Computers and Mathematics with Applications, 28(5), pp: [2] I.H. Abdel-Halim Hassan (2008) Application to differential transformation method for solving systems of differential equations, Applied Mathematical Modelling, 32(12), pp: [3] G. Adomian (1988) A review of the decomposition method in applied mathematics, Journal of Mathematical Analysis and Applications, 135, pp: [4] G. Adomian (1990) A review of the decomposition method and some recent results for nonlinear equations, Mathematical and Computer Modelling, 13(7), pp: [5] G. Adomian and R. Rach (1992) Noise terms in decomposition solution series, Computers and Mathematics with Applications, 24(11), pp: [6] G. Adomian (1994) Solving Frontier problems of Physics: The decomposition method, (Kluwer Academic Publishers). [7] Y. Cherruault (1990) Convergence of Adomian s method, Mathematical and Computer Modelling, 14, pp: [8] Georgii Evg. Pukhov (1982) Differential Transforms and Circuit Theory, Circuit Theory and Applications, 10, pp: [9] M.J. Jang, C.L. Chen,Y.C. Liy (2000) On solving the initial value problems using the differential transformation method, Applied Mathematics and Computation, 115, pp: [10] M.J. Jang, C.L. Chen, Y.C. Liy (2001) Two-dimensional differential transform for Partial differential equations, Applied Mathematics and Computation, 121, pp: [11] Md. Jahangir Hossain, Md. Shah Alam, Md. Babul Hossain (2017) A study on numerical solutions of second order initial value problems (IVP) for Ordinary Differential Equations with fourth order and Butcher s fifth order Runge-Kutta methods, American Journal of Computation and Applied Mathematics, 7(5), pp: [12] Jian-Lin Li (2009) Adomian s decomposition method and homotopy perturbation method in solving nonlinear equations, Journal of Computational and Applied Mathematics, 228, pp: [13] Juddy H. Arias, Hector J. Martinez, Lilian S. Sepulveda and Olga Vasilieva (2015) Predator-Prey model for analysis of Aedes Aegypti population dynamics in Cali, Columbia, International Journal of Pure and Applied Mathematics, 105(4), pp: [14] Malavige G. N., Fernando, S., Fernando, D. J. and Seneviratne, V. L. (2004) Dengue viral infections, Postgraduate Medical Journal, 80, pp:

10 [15] Mirzaee F. (2011) Differential transform method for solving linear and nonlinear systems of ordinary differential equations, Applied Mathematical Sciences, 5(70), pp: [16] A. S. Oke (2017) Convergence of Differential Transform Method for Ordinary Differential Equations, Journal of Advances in Mathematics and Computer Science, 24(6), pp:1 17. [17] Suarez-Rubio M, and Suarez M. F (2004) The use of the copepod Mesocyclops longisetus as a biological control agent for Aedes aegypti in Cali, Colombia., Journal of the American Mosquito ControlAssociation, 20(4), pp: [18] Yang H, Macoris M, Galvani K, Andrighetti M and Wanderley D (2009) Assessing the effects of temperature on the population of Aedes aegypti the vector of dengue, Epidemiology and Infection, 137, pp: [19] J.K. Zhou (1986) Differential Transformation and Its Applications for Electrical Circuits, (Huazhong University Press, Wuhan, China). (in Chinese) 13858

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