An Extended Kalman Filter Implementation for ...

1

An Extended Kalman Filter Implementation for Filariasis Transmission Estimation

Ais Maulidia Maziyah, Mochamad Isman Safii, Prof. Dr. Basuki Widodo, M.Sc Matematika, Fakultas Matematika dan Ilmu Pengetahuan Alam, Institut Teknologi Sepuluh Nopember Jl. Arief Rahman Hakim, Surabaya 60111 Indonesia Abstract— Filariasis is a spread disease that is caused by filaria worm transmitted by several kind of mosquitos. The filariasis transmission involving human and mosquitos. On that case, analysis of filariasis transmission model is did by medication process. On this model, will estimating the spreading using Extended Kalman Filter. Extended Kalman Filter is an expansion of Kalman Filter Methode that can be used for estimating the model of nonlinear system and continu system. First we do model discretization of time. The final result of this estimation is a simulation using Matlab Key Words — Filariasis Transmission Model Using Meditication, Extended Kalman Filter I. INTRODUCTION

Around 20% of world population or 1.1 billion population in 83 countries are estimated to be infected by Filariasis, moreover in the tropical area and some in the subtropical area. This disease causes disability, social stigma, psichologycal problem, and decreasing of the work productivity of the patients, family, and the sociaty, hence causes a big loss [1]. Based on the result of Filariasis quick survey in Indonesia 42% of 7000 questionairs on 2000s, the amount of chronical patients were 6233 people spread in the 674 hospitals, 1553 village in 231 districts, in 26provinces. This data could not describe the real condition, because it was only reported by 42% of 7221 hospitals. The Filariasis endemicity rate since 1999s is still so high where the prevalence of microfilaremia is 3.1% [1]. From public health office of east java government data, known that on 2012s where the citizen was about 38.052.950 people and 341 Filariasis patients were treated on the hospitals around east java. At 2013s the total of east java citizen was about 38.318.719 people and the Filariasis patient was about 359 people. Untill at 2013s, the biggest number of filariasis patient in east java was in the Lamongan district. Where the patents were 56 people of 1.200.558 Lamongan citizen. Filariasis Transmisi in an area can be formulated in the

mathematic modelling. That mathematic modelling can be used for estimating the spreading of filariasis on some areas by Extended Kalman Filter. The result of the estimation is expected can be used for generating the countermeasures and preventing the Filariasis transmission in an area. I. Problem Statement 1. How’s the way to determine the stability of every endemic and free-desease equililbrium point? 2. How is the analysis result interpretation of filariasis transmission model with the treatment process and its simulation? 3. How is the estimation of Filariasis transmission using Extended Kalman Filter? 4. How is the simulation of the filariasis transmission estimation result using Extended Kalman Filter? II. Mathematical Modelling There are some definitions in the Filariasis transmission model :  is the population of the healthy human whom are suspectible by Filiaris  is the population of infected people without clinical symptom and can transmit the desease  is the population of the chronical disability  is the population of healthy mosquitos that are suspectible of filaria transmission  is the population of infected mosquitos From the definitions above, obtains a model compartement diagram of Filariasis transmission with medication below:

2

That model can be written as: (1) (2) (3) (4) (5) ( )

( ) ( )

And will be obtained the equilibrium point of the free disease : ( )

( )

(

)

( )

The number population of total human is constant, so can be written as , where

(

) (

Since population rate

and hence the equation of and will be reduced, by subtituting onto equation (2) and equation (1) is

erased and substituting onto eequation (5) and equation (4) will be erased, hence will get the new equation from the model of reduction result as below:

)

(

)

(6)

And the amount of population of the mosquito is also constant, so can be defined as , where

(7) (

(

) (

(

)

The equilibrium point of disease free can be obtained b y counting . The condition of disease free can be happened whenthe population of so, will be derived the equilibrium point of the disease free. On the equation (1) and (4) where so,

)

)

(8)

III. Analytical / Numerical Solution Included Description  Extended Kalman Filter Implementation Discretization is did by using foward finit different for the variable changing towards the time. Discretization is did onto equation (1) - (5) so that obtains this equations : (

()

(

)

( )

( ))

(

() ()

( )

(10) ) (11)

( (

3

)

)

(

)

(

(12) ( ))

()

( )

(13) (

( ))

( )

(14)

( )

Assuming that:

Can be written as:

Tabel 2 parameters NO Parameter Nilai Parameter 1 235 2 0.014 3 0.001 4 243 5 0.2 6 45000 7 12.67 8 0.5 10 500 11 0.9 12 87 13 0.001 The simulation result and value of RMSE by using the parameter and initial value based on the tabel 1 and 2 with 70 iterations obtain the graph with the computation time 3,183017 is below:

[ ] ()

( (

() ()

()

()

()

()

(

( ))

() ()

(

() ()

()

( ))

()

()

( [

( ))

()

()

() ( )) ( ))

() ()

]

Defined the state space : Picture 1 Comparison Graph between Real Value and Estimation Value of Healthy People Population( ) The red color on the graph determine the real value that contains noise, however the blue color determine the value of estimation result. Picture 1 is the comparison between the real value and the value of estimation result of the healthy human population. The result shows that the difference between both is 0.74

[ ] The system model : (

)

The measurement parameter :

Then implementing the Extended Kalman Filter algorithm onto the equation (14) – (18)  Extended Kalman Filter Simulation Result In this simulation, the initial value and parameter that will be used is stated below: Tabel 1 Inital point of eacg populations No Initial Value population when 1 2 3 4 5

80

15

Picture 2 Comparison Graph between Real Value and Estimation Value of Infected People Population (A) Picture 2 is the comparisson between the real time

4

with the value of estimation result of the suspectable human. The result shows that the difference between both is 0.07

Picture 5 Comparison Graph between Real Value and Estimation Value of Infected Mosquitos population ( )

Picture 3 Comparison Graph between Real Value and Estimation Value of Chronical Disabiliy People Population (K) Picture 3 is the comparisson between the real time with the value of estimation result of the chronical disability people. The result shows that the difference between the real value that contaons noise and the estimation value is 0.77

Picture 5 is the comparisson between the real time with the value of estimation result of the suspectible mosquitos population. The result shows that the difference between the real value that contaons noise and the estimation value is 0.83

Picture 6 Error Graph between Real Value and Estimation Value of All Populations Picture 4 Comparison Graph between Real Value and Estimation Value of Healthy Mosquitos Population ( )

Picture 4 is the comparisson between the real time with the value of estimation result of the health mosquitos population. The result shows that the difference between the real value that contaons noise and the estimation value is 0.2

Picture 6 shows the graph of the error of the real value and estimation result value among all of populations. Show that the smallest error value is on the suspectible people population On the picture 1-2 show that the graph of the suspectible people population estimation result is more approach the real graph than another population estimation. Proven by RMSE value of the smallest suspectible human population, is [0 1 0 0 0], describe that the measurement data that is used is the infected people population. Tabel 3 RMSE AverageValue RMSE Average Value Iterations 70

0.7414

0.0728

0.7725

0.2042

0.8377

On the 3 shows that the RMSE value of every

population is relative small, that is the errir value (ne) on the interval 0.07 < ne < 0.83. Hence, overall this case can be said that the Extended Kalman Filter method is suitable for estimating tge Filariasis disease transmission. V. Conclusion 1. Filariasis transmission model that already examined,obtain the equilibrium point and stability analysis as below: a. Equilibrium point of free-disease : (

( )

(

)

)

Local asymtotically stable, is fulfilled when ,

,

b. Second condissiom endemic equilibrium point (

( )

)

where

( )

√

Third condission : (

( )

)

Where:

( )

√ With assumption from

Local asymptotically stable is fulfilled when , ,

2. Extended Kalman Filter that is used can be applied for estimating filariasis transmission. This case is based on the known Error RMS is relatively small on each state, that is the error value (ne) is between 0.07