MIV application for Sir Mode from Spread of Dengue Fever Diseas

550 Satisfied. So q ascending from 0 to 1, the completion of various ߶ ௜ ݐ; ݍ of the initial assessment ݖ ௜,଴ to completion ݖ ௜ ݐ. Expantion ߶ ௜ ݐ; ݍ by a Taylor series against q, obtained : ߶ ௜ ݐ; ݍ = ݖ ௜,଴ ݐ + ෍ ݖ ௜,௠ ݐݍ ௠ ା∞ ௠ୀଵ , 10 where ݖ ௜,௠ = 1 ݉ ߲ ௠ ߶ ௜ ݐ; ݍ ߲ݍ ௠ ฬ ௤ୀ଴ . 11 If additional linear operator, the initial guess of additional parameters ℏ ௜ ,, and additional function is selected, then the equation of the series 10 converges to q = 1 and ߶ ௜ ݐ; 1 = ݖ ௜,଴ ݐ + ෍ ݖ ௜,௠ ݐ, ା∞ ௠ୀଵ 12 This became the settlement of non linear equations of origin, as proved by Liao. As with ℏ = −1, equation 8 becomes: 1 − ݍܮൣ߶ ௜ ݐ; ݍ − ݖ ௜,଴ ݐ൧ + ݍܰ ௜ ൣ߶ ௜ ݐ; ݍ − ݖ ௜,଴ ݐ൧ = 0, 13 Which method mostly used in the HPM. He, 1999. From 13, the construction of the equation can be deduced from the zero- order deformation equation. Defined vector ݖ̅ ௜,௡ = ൣݖ ௜,଴ ݐ, ݖ ௜,ଵ ݐ, ⋯ , ݖ ௜,௡ ݐ൧ 14 Differentiation 25 m times as much on the setting of parameters q and then let q = 0 and finally divided them by m, We call to-order deformation equation m. ܮൣݖ ௜,௠ ݐ − ߯ ௠ ݖ ௜,௠ ݐ൧ = ℏ ௜ ܴ ௜,௠ ൫ݖ̅ ௜,௠ିଵ ൯, 15 where ܴ ௜,௠ ൫ݖ̅ ௜,௠ିଵ ൯ = 1 ݉ − 1 ߲ ௠ିଵ ߶ ௜ ݐ; ݍ ߲ݍ ௠ିଵ ቤ ௤ୀ଴ , 16 and ߯ ௠ = ቄ 0, ݉ ≤ 1, 1, ݉ 1. 17 It is emphasized that ݖ ௜,௠ ݐ ݉ ≥ constructed by the linear equation 15 with linear boundary conditions that com from original problem, which can be easil solved with a computer symbol cra MAPLE or MATHEMATICA.

3.1. MIV application for Sir Mode from Spread of Dengue Fever Diseas

Variation iteration method is a method o semi analytic powerful and widely used solve differential equations. The bas theory for using MIV, first, rewritten SIR model of dengue fever as follows:   1 h dx x t x t z t dt      dy x t z t y t dt       1 dz z t y t z t dt      where ߤ ௛ is human life span, the shape o the number susceptible rate ܣ, ܾߚ ௛ is th number of the host population, ߤ ௩ is th life span of mosquitoes, ܰ ௛ is the numb of the host population, ߙ = ௕ఉ ೓ ஺ ఓ ೡ ே ೓ , ߚ ߛ ௛ + ߤ ௛ , ߛ = ܾߚ ௩ and ߜ = ߤ inhomogeneous form To use the SIR model MIV, built function justification as follows: ݔ ௡ାଵ ݐ = ݔ ௡ + න ߣ ଵ ݏ ൤ ݀ݔ ௡ ݀ݏ ௧ ଴ − ߤ ௛ 1 − ݔ ௡ + ߙݔ ௡ ݖ ௡ ෧൨ ݀ݏ, 18 ݕ ௡ାଵ ݐ = ݕ ௡ + න ߣ ଶ ݏ ൤ ݀ݕ ௡ ݀ݏ ௧ ଴ − ߙݔ ௡ ݖ ௡ ෧ + ߚݕ ௡ ൨ ݀ݏ, 19 551 ݖ ௡ାଵ ݐ = ݖ ௡ + න ߣ ଷ ݏ ൤ ݀ݖ ௡ ݀ݏ ௧ ଴ − ߛݕ ௡ + ߛݕ ௡ ݖ ௡ ෧ + ߜ ଵ ݖ ௡ ൨ ݀ݏ, 20 Where ߣ ௜ , ݅ = 1,2,3 is a general Lagrange coefficient which can be identified optimally by theory of variation and the subscript n denotes to-n. To obtain ߣݏ is optimal, do the following: ߜݔ ௡ାଵ = ߜݔ ௡ + න ߜߣ ଵ ݏ ൤ ݀ݔ ௡ ݀ݏ ௧ ଴ − ߤ ௛ 1 − ݔ ௡ + ߙݔ ௡ ݖ ௡ ෧൨ ݀ݏ, 21 ߜݕ ௡ାଵ = ݕ ௡ + න ߜߣ ଶ ݏ ൤ ݀ݕ ௡ ݀ݏ ௧ ଴ − ߙܿߚݕ ௡ ෦൨ ݀ݏ, 22 ݖ ௡ାଵ = ݖ ௡ + න ߣ ଷ ݏ ൤ ݀ݖ ௡ ݀ݏ − ߛݕ ௡ ෦ ௧ ଴ + ߛݕ ௡ ݖ ௡ ෧ + ߜ ଵ ݖ ௡ ෦൨ ݀ݏ, 23 where ݔ෤ ௡ , ݕ෤ ௡ , ݕ ௡ ݖ ௡ ෧ , ݔ ௡ ݖ ௡ ෧, and ݖǁ ௡ considered as a discontinuous variation i, ݔ෤ ௡ , ݕ෤ ௡ = 0 and ݖǁ ௡ = 0. Then, we have ߜݔ ௡ାଵ = ߜݔ ௡ + න ߜߣ ଵ ݏ ൤ ݀ݔ ௡ ݀ݏ ௧ ଴ + ߤ ௛ ݔ ௡ ൨ ݀ݏ, 24 ߜݕ ௡ାଵ = ݕ ௡ + න ߜߣ ଶ ݏ ൤ ݀ݕ ௡ ݀ݏ ௧ ଴ + ߚݕ ௡ ൨ ݀ݏ, 25 ߜݖ ௡ାଵ = ߜݖ ௡ + න ߜߣ ଷ ݏ ൤ ݀ݖ ௡ ݀ݏ ௧ ଴ + ߜ ଵ ݖ ௡ ൨ ݀ݏ, 26 or ߜݔ ௡ାଵ = ߜݔ ௡ + න ൤ߜߣ ଵ ݀ݔ ௡ ݀ݏ ௧ ଴ + ߜߣ ଵ ߤ ௛ ݔ ௡ ൨ ݀ݏ 27 ߜݕ ௡ାଵ = ߜݕ ௡ + න ൤ߜߣ ଶ ݏ ݀ݕ ௡ ݀ݏ ௧ ଴ + ߜߣ ଶ ߚݕ ௡ ൨ ݀ݏ 28 ߜݖ ௡ାଵ = ߜݖ ௡ + න ൤ߜߣ ଷ ݀ݖ ௡ ݀ݏ ௧ ଴ + ߜߜ ଵ ߣ ଷ ݖ ௡ ൨ ݀ݏ 29 then ߜݔ ௡ାଵ = ߜ1 + ߣ ଵ ݔ ௡ + න ߜ[ߣ′ ଵ ௧ ଴ + ߣ ଵ ߤ ௛ ] ݔ ௡ ݀ݏ, 30 ߜݕ ௡ାଵ = ߜ1 + ߣ ଶ ݕ ௡ + න ߜ[ߣ′ ଶ ௧ ଴ + ߣ ଶ ߚ]ݕ ௡ ݀ݏ, 31 ߜݖ ௡ାଵ = ߜ1 + ߣ ଷ ݖ ௡ + න ߜ[ߣ′ ଷ ௧ ଴ + ߜߜ ଵ ߣ ଷ ] ݖ ௡ ݀ݏ, 32 Thus , the following stationary condition obtained : ߜݔ ௡ : 1 − ߣ ଵ ݐ| ௦ୀ௧ = 0, ߜݕ ௡ : 1 − ߣ ଶ ݐ| ௦ୀ௧ = 0, ߜݖ ௡ : 1 − ߣ ଷ ݐ| ௦ୀ௧ = 0, ߜݔ ௡ ′ : ߣ ଵ ′ ݏ + ߤ ௛ ߣ ଵ ݏห ௦ୀ௧ = 0, ߜݕ ௡ ′ : ߣ ଶ ݏ + ߚߣ ଶ ݏ| ௦ୀ௧ = 0, ߜݖ ௡ ′ : ߣ ଷ ݏ + ߜ ଵ ߣ ଷ ݏ| ௦ୀ௧ = 0, Solution of the system of equation obtained: ߣ ଵ ݏ = −݁ ఓ ೓ ௦ି௧ , ߣ ଶ ݏ = −݁ ఉ௦ି௧ , ߣ ଷ ݏ = −݁ ఋ భ ௦ି௧ , 33 Here, the general Lagrange coefficients 33 is described by the Taylor series an selected for only one term in th calculation easier, general Lagrang coefficients can be written as follows: ߣ ଵ ݏ = −1, ߣ ଶ ݏ = −1, ߣ ଷ ݏ = −1, 34 552 Substitution of the general Lagrange multipliers in 34 to equation 18-20 yields the following iteration formula: ݔ ௡ାଵ ݐ = ݔ ௡ − න ൤ ݀ݔ ௡ ݀ݏ ௧ ଴ − ߤ ௛ 1 − ݔ ௡ + ߙݔ ௡ ݖ ௡ ൨ ݀ݏ, 35 ݕ ௡ାଵ ݐ = ݕ ௡ − න ൤ ݀ݕ ௡ ݀ݏ − ߙݔ ௡ ݖ ௡ ௧ ଴ + ߚݕ ௡ ൨ ݀ݏ, 36 ݖ ௡ାଵ ݐ = ݖ ௡ − න ൤ ݀ݖ ௡ ݀ݏ ௧ ଴ − ߛݕ ௡ + ߛݕ ௡ ݖ ௡ + ߜ ଵ ݖ ௡ ൨ ݀ݏ. 37 Iteration begins with the initial approximation as obtained from the data of the Indonesian health minister 2007, ܿ ଵ = ଻଺଻ହସ଴଺ ଻଺଻ହ଼ଽଷ , ܿ ଶ = ସ଼଻ ଻଺଻ହ଼ଽଷ , ܿ ଷ = 0.056, ߙ = 0.232198, ߚ = 0.328879, ߛ = 0.375, and ߜ ଵ = 0.0323. The iteration formulas 87 - 89 are obtained: ݔ ଵ = 0.9999365546 − 0.0130022687 ݐ, 38 ݕ ଵ = 0.00006344538675 + 0.01298140513 ݐ 39 ݖ ଵ = 0.056 − 0.001786340333 ݐ, 40 ݔ ଶ = 0.9999365546 − 0.0130022687 ݐ + 2.922132174 × 10 ିସ ݐ ଶ −1.797714851 × 10 ି଺ ݐ ଷ , 41 ݕ ଶ = 0.00006344538675 + 0.01298140513 ݐ + 0.1797714851 × 10 ିହ ݐ ଷ −0.002426569924 ݐ ଶ , 42 ݖ ଶ = 0.056 − 0.001786340333 ݐ + 0.002326579355 ݐ ଶ +0.2898650945 × 10 ିହ ݐ ଷ , 43 ݔ ଷ = 0.9999365546 − 0.01300226807ݐ − 0.0001831331308ݐ ଷ +0.0002922132174 ݐ ଶ + 0.1728532016 × 10 ିଵଶ ݐ ଻ +0.1290829001 × 10 ିଽ ݐ ଺ − 0.2997118573 × 10 ି଻ ݐ ହ +0.1623956764 × 10 ିହ ݐ ସ , 44 ݕ ଷ = 0.00006344538675 + 0.01298140513 ݐ + 0.000449144614 ݐ ଷ −0.00242656993 ݐ ଶ − 0.1728532016 × 10 ିଵଶ ݐ ଻ −0.1290829001 × 10 ିଽ ݐ ଺ + 0.2997118573 × 10 ି଻ ݐ ହ −0.1771743755 × 10 ିହ ݐ ସ , 45 ݖ ଷ = 0.056 − 0.00178634033 ݐ − 0.000308504557 ݐ ଷ +0.002326579355 ݐ ଶ − 0.2791579206 × 10 ିଵଶ ݐ ଻ +0.1782033109 × 10 ିଽ ݐ ଺ + 0.4208392710 × 10 ି଺ ݐ ଷ −0.3102165044 × 10 ିହ ݐ ସ , and so on. 46

3.2. MAH Applications for SIR Mod of the Spread of Dengue Feve