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