TELKOMNIKA ISSN: 1693-6930
A Cellular Automata Modeling for Visualizing and Predicting Spreading… Puspa Eosina 233
1 1
1 1
1 1
1 1
mod 4
i i
i i
i i
j j
j j
j j
x x
x x
x x
6 Where mod 4 indicates the number of states that
i j
x
might describe the j
th
cell in the i
th
-row which can take on the value 0,1,2, and 3. The values of the cells in the first row, which range
from 0 to 3, were randomly assigned by Simple Linear Congruential Generators.
2.5. Verification and Validation
Evaluation of the proposed models was conducted by doing verification and validation. Model verification aims to ensure that the CA model has been implemented correctly. Moreover,
the purpose of validation is to determine whether the theory and assumptions underlying the preparation of this model are correct [24]. The SIR-model Susceptible-Infected-Recovered is a
simple mathematical model based on ODE that has been proven to be an acceptable model in the epidemic fields [25]. The SIR model was represented as shown:
7 8
9 Where
S = number of susceptible, I = number of infectious, and R = number of recovered.
Case represents the transmission probability of the disease and represents the period of
infection. Verification of the model was conducted by comparing the tendency of graphs yielded
by the proposed model, and the trend graphs yielded by the SIR model. Sequentially, validation of the model was performing by measuring the proximity of the simulation results of the
proposed model and those of the SIR-model using a correlation coefficient measure to compute similarity [26]:
1
, 1
n i
i i
X Y
X X Y
Y Corr X Y
n
10 Where
i
X
and
X
are declared time-series data and the average generated by the proposed model, respectively,
i
Y
and
Y
are declared time-series data and the average generated by the SIR model, respectively.
X
and
Y
represented standard deviation of variable X and variable Y. The similarity value lies between 0.5 - 1. The closer it gets to 1, the two time-series data can
be said to be similar [27]. 3. Results and Analysis
3.1. Probabilistic Functions Obtained as Rule on CA Model
The spreading pattern and prediction model were represented by the probabilistic function that represents the CA rule. The probabilistic function obtained in this research is
described as follows:
4 4
1 i 1
1
| S .
S |
max
n n
n j
i i
j
f P V V
P V V
11
dS S
I dt
dI
S I
I dt
dR I
dt
ISSN: 1693-6930
TELKOMNIKA Vol. 14, No. 1, March 2016 : 228 – 237
234 Where
n
V
represents state value of cell V in n-th period, and
1
|
n n
o i
P V S V
is the probability of V
that is at S
i
in the next period. This function allows us to choose the maximum value of the probability of the state change. It means that the change of a state in the cell of V
in the next period depends on the maximum probability value obtained from equation 12-15. These probabilities consist of four possibility values that are defined as follows.
4 1
1 1
2 3
4 1
1 1
S | ,
, ,
| S .
S |
n n
n n
i i
P V V V V V
P V V P V
V
12
4 1
2 1
2 3
4 2
2 1
S | ,
, ,
| S .
S |
n n
n n
i i
P V V V
V V P V
V P V
V
13
4 1
3 1
2 3
4 3
3 1
S | ,
, ,
| S .
S |
n n
n n
i i
P V V V
V V P V
V P V
V
14
4 1
4 1
2 3
4 4
4 1
S | ,
, ,
| S .
S |
n n
n n
i i
P V V V
V V P V
V P V
V
15\
The values of
1
| S
n i
P V V
;
2
| S
n i
P V V
;
3
| S
n i
P V V
;
4
| S
n i
P V V
were obtained from the Emission Probabilities Matrix using the equation 15-18. Moreover, the values of
1 1
S |
n n
P V V
;
1 2
S |
n n
P V V
;
1 3
S |
n n
P V V
;
1 4
S |
n n
P V V
were obtained from T,
1
|
n n
P V V
was obtained based on the formula described in Table 3 with the results of probability values as follows Equation 16:
1
0.60 0.32
0.06 0.01
0.54 0.30
0.13 0.03
0.59 0.24
0.18 0.00
0.60 0.20
0.20 0.00
|
n n
P V V
16
This matrix shows that the change from state S
4
to S
1
has the highest probability value. The matrix also shows that the possibility of a cells state change next period from S
3
to S
4
was very small or never occurred. Moreover, if the condition of a cell was in the state of S
4
, the state tends to change to the better condition because the probability of the cell’s state to keep its
state was very small or never occurred. It means that there were always the preventive actions to stop the spreading of Dengue Fever diseases in this area.
In this research, E is a matrix for representing the state change probabilities of a certain area affected by its neighborhood. There were four matrices E as sequently, from Equation 17
up to Equation 20 as follows.
1
0.6579 0.2500
0.0789 0.0132
0.4423 0.4038
0.1346 0.0192
0.5833 0.2500
0.1667 0.0000
0.0000 0.5000
0.2500 0.2500
| P V
V
17
Equation 17 described the probability of a state change of cell V
1
on the next period that is affected by the state change of cell V
. It was shown that the probability of V
1
’s state changing to S
4
was very small or never occurred, while V was in S
3
. Moreover, this matrix also showed that the change from the state of area V
1
to S
1
, while V was in S
4
never occurred.
TELKOMNIKA ISSN: 1693-6930
A Cellular Automata Modeling for Visualizing and Predicting Spreading… Puspa Eosina 235
2
0.6538 0.2692
0.0769 0.0000
0.4255 0.4255
0.1277 0.0213
0.4667 0.2667
0.2000 0.0667
0.2500 0.5000
0.2500 0.0000
| P V
V
18
Equation 18 shows that the state change of V
2
to S
4
, affected by V , would never occur while
the state of V is in the condition of S
1
or S
4
.
3
0.6250 0.2875
0.0875 0.0000
0.4222 0.4667
0.0667 0.0444
0.3750 0.4375
0.1250 0.0625
0.3333 0.3333
0.0000 0.3333
| P V
V
19
Equation 19 describes the probability of a state change of cell V
3
on the next period that is affected by the state condition of V
. Equation 20 describes the probability of a state change of cell V
4
on the next period that is affected by the V state condition. From the four matrices
above, it is concluded that the extreme change conditions of the neighborhoods to S
4
, affected by the focus area, are very rare.
4
0.6456 0.2532
0.0886 0.0127
0.4468 0.4255
0.0851 0.0426
0.3750 0.3750
0.1875 0.0625
0.0000 0.5000
0.5000 0.0000
| P V
V
20
3.2. Prediction Model Results Obtained Using CA Model
