Technical Report (Comparison Of Dengue Cases For Chosen District In Selangor By Using Fourier Series)

ACKNOWLEDGEMENTS

IN THE NAME OF ALLAH, THE MOST GRACIOUS, THE MOST MERCIFUL

Firstly, I am grateful to Allah S.W.T for giving me the strength to complete this project
successfully. I would like to express my gratitude Associated Professor Maheran
Nuruddin for the guidance and support to this report.

Special thanks to my family, especially my parent who always support and pray for my
success. I wish to thank my friends for their support. They have been very supportive
throughout the completion of this project.

Last but not least, thank you again for those who spent time and effort with me in
completing this report, directly or indirectly. Without Allah bless and the all kindness of
these people, I will never succeed in completing this project. Thank you so much.
Wasalam.

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ................................................................................................. i
TABLE OF CONTENTS .................................................................................................... ii
LIST OF TABLES ............................................................................................................. iii
LIST OF FIGURES ........................................................................................................... iii
ABSTRACT ....................................................................................................................... iv
1. INTRODUCTION ........................................................................................................ 1
2. METHODOLOGY ....................................................................................................... 4
3. IMPLEMENTATION ................................................................................................... 6
4. RESULTS AND DISCUSSION ................................................................................. 32
5. CONCLUSIONS AND RECOMMENDATIONS ..................................................... 33
REFERENCES ................................................................................................................. 34

ii

LIST OF TABLES
Table 1. Fourier series calculation by using excel for dengue cases in Shah Alam (2009) . 8
Table 2. Fourier series calculation by using excel for dengue cases in Gombak (2009) ... 12
Table 3. Fourier series calculations by using excel for dengue cases in Klang (2009) ..... 16
Table 4. Fourier series calculations for dengue cases in Shah Alam (2010) ..................... 20
Table 5. Fourier series calculations by using excel for dengue cases in Gombak (2010) . 24
Table 6. Fourier series calculations by using excel for dengue cases in Klang (2010) ..... 28
Table 7. Fourier series equations on 1st harmonic for 2009 .............................................. 32
Table 8. Fourier series equations on 1st harmonic for 2010 .............................................. 32
Table 9. Analysis from graph using maple software for 2009 ........................................... 32
Table 10. Analysis from graph using maple software for 2010 ......................................... 32

LIST OF FIGURES

Figure 1. Graph of dengue cases in Shah Alam, Gombak and Klang for year 2009 ........... 6
Figure 2. Graph of dengue cases in Shah Alam, Gombak and Klang for year 2010 ........... 7
Figure 3. Fourier series graph plotted for dengue cases in Shah Alam (2009) .................. 11
Figure 4. Fourier series graph plotted for dengue cases in Gombak (2009) ...................... 15
Figure 5. Fourier series graph plotted for dengue cases in Klang (2009) .......................... 19
Figure 6. Fourier series graph plotted for dengue cases in Shah Alam (2010) .................. 23
Figure 7. Fourier series graph plotted for dengue cases in Gombak (2010) ...................... 27
Figure 8. Fourier series graph plotted for dengue cases in Klang (2010) .......................... 31

iii

ABSTRACT

Dengue is the most dangerous mosquito virus infection to the human in the world. Up to
100 million cases are reported annually and some two billion people are at risk of
infection in the world. There is no specific cure or medicine to shorten the course of
dengue. The occurrence of dengue in Malaysia has become more serious year to year.
The aims of this study are to know the pattern of dengue cases that happened in chosen
district and to obtain the highest point for dengue cases in chosen district by referring to
Fourier series graph plotted. This project focuses on certain districts which had
recorded the highest dengue cases among district in Malaysia which are Shah Alam,
Gombak and Klang. It is difficult to determine and predict the dengue cases for the next
year by following the trend line that is generated by Excel. Thus, the alternative that we
have is to transform the graph into a periodic graph using Fourier series, so that the
highest point for the dengue cases can be determined. Fourier series is an expansion of a
periodic function of period which the base is the set of sine functions. Hence, Fourier
series is one of the alternative methods to compare and explain the pattern of dengue
cases recorded. The result between year 2009 and 2010 show the number of dengue cases
seasonally peak at first quarter of year which averagely recorded in period week 7 to
week 14 (February to April).

iv

1. INTRODUCTION

Dengue is the most dangerous mosquito virus infection to the human in the world. Ang
and Li (1999) stated that up to 100 million cases are reported annually and some two
billion people are at risk of infection in the world. Dengue viruses are transmitted from
vector (mosquitoes) to the susceptible human beings by various mosquitoes such as
Aedes aegypti and Aedes albopictus. From that, the infected person will have a few
symptoms such as high fever (40°C), chills, headache, pain in the eyes, deep muscle and
joint pains and extreme fatigue. Actually, the infected person will have high fever for two
to four days. Then, the body temperature will drop rapidly and intense sweating takes
places. But, patient’s body will show up small red bumps. These are a few symptoms that
will happen to the infected person. If the patient does not take immediately treatment,
dengue may cause death.

Knowing how dengue being transmitted is very important. This is relevant to this study
because we must identify and know who is the vector and the host. Basically, dengue
viruses are transmitted from the vector (mosquitoes) to the host (humans). The
transmitted dengue virus process happened by mosquitoes bite during mosquitoes blood
feeding. The mosquitoes also may carry the virus from one host to another host. When
the virus has been transmitted to the host (humans) incubation period will occur. The
dengue viruses multiply during incubation time. After three to five days, the symptoms
of dengue will appear and attack patients.

There is no specific cure or medicine to shorten the course of dengue. Actually, the
medicine provided by doctors is to reduce and alleviate the symptoms and sign of dengue.
In this situation, the patient (infected person) takes paracetamol to relieve muscle and
joint aches, fever and headache. The patient is advice to keep rest in a screened room to
prevent mosquitoes from entering. The dengue virus will be transmitted to another host
(human) if the patient is bitten second times. After this treatment, in a few days, we can
define the patient is fully recovered and in the best condition (recover person) when the
symptoms had disappeared.

The occurrence of dengue in Malaysia had become more serious year to year. The
Ministry of Health Malaysia (2009) stated that dengue has become pandemic. Besides
that, people did not take this problem as a serious problem. In order to increase the
people’s sensitivity of dengue, the Ministry of Health has done many activities and
campaign such as advertisement through the television and internet. The activities and
campaign also include involvement of students in primary and secondary schools. For
example, the competition “AntiAedes Ranjer Ridsect” organized by Sara Lee Company
(Ridsect) which cooperated with Ministry of Health Malaysia.

1

In order to analyze the dengue cases which happened in Malaysia for this study, Fourier
series was choose because of its availability to present and show the new perspective
analysis of dengue cases. Zill and Cullen (2009) stated that the representation of a
function in the form of a series is widely and frequently used to solve and explain the
common problem situation.

The history of Fourier series started when Bernoulli, D’ Allembert and Euler (1750) had
used and introduced the idea of expanding a function in the form a series to solve the
associated with the vibration of strings. Then, Joseph Fourier who a French physicist,
(1768-1830) improved and developed the approach of Fourier series where it was
generally useful nowadays. However, the search had done by Joseph Fourier gave impact
to all mathematicians and physicists at that time such as Laplace, Poisson and Lagrange.
They doubt and debate about Fourier’s work because it opposite and inversed to their
idea. But, the text of Joseph Forier which Theorie Analytique de la Chaleur (The
Analytical Theory of Heat) become the source for the modern method in order to solve
problems associated with partial differential equations subject to prescribed boundary
conditions.

Nowadays, the application of Fourier series analysis is commonly used in physic and
electrical engineering sector which how frequency associated to a dynamical systems.
The text from Joseph Fourier influenced in created electrical component such as
electronic rectifiers. Fourier series also is the best method to analyze the data series such
as dengue cases which useful to compare and determine the dengue cases happened in
Malaysia.

Angove (2009) analyze the periodic time domain voltage waveform and convert it to the
frequency domain which always uses in electronic communication systems. For example
a waveform usually decomposed into sum of harmonically related sine, cosine waveform
and constant which is known as Fourier series.

Klingenberg (2005) showed the way to apply and calculate Fourier series analysis by
using Microsoft Excel. Excel generally shows the magnitude versus time is known
waveform. Klingenberg (2005) done the experiments call for the “harmonic content” of a
reproduced waveform is a display of the magnitude of the waveform (Y-axis) versus the
frequency (X-axis). In other word, we called it as frequency spectrum and it allows
visualizing a waveform according to its frequency content.

Kvernadzi, Hagstrom and Shapiro (1999) studied about the utilization of the truncated
Fourier series and it applies as a tool for the approximation of the points of discontinuities
and the magnitudes by using integrals. Abas, Daud and Yusuf (2009) studied about
rainfall by using Fourier series with significant number of harmonics is fitted to the
model’s parameter. The results of their studies showed that statistical properties of the
estimated rainfall series were able to match most of those of the historical series. The
Fourier series makes the model more parsimony by grabs the seasonal fluctuations within
the model.

2

The strategy or plan must be systematic. So, modeling how dengue spread among
population localized in a district guides the Ministry of Health Malaysia to prevent these
epidemics become more danger to community. The model were showed the seasonal
pattern that are useful in prevent in a spread of dengue. Favier (2006) suggest that,
statistical analyses of longitudinal surveys sites are needed before choose the right
parameters.

The scope of this study was in small scale because Favier, Degallier and Dubois (2005)
stated that possibility of transmission dengue also depends on the population density and
previous immunization. Sometimes, factor likes rainfalls, temperature must be
considered. The virus progression occurs at a daily scale; therefore it must be recorded in
weeks or days to be more accurate and precise. So, the prediction and modeling of
dengue repartition and dynamics raises must different depends on the situation and place.
Since the scope of study is in small scale, the result will be more accurate. For instance,
modeling of dengue prevalence is conceivable at town-scale like Shah Alam, Subang and
Klang but not at global scale, where long-range interactions cannot model accurately.

Dengue will impact high death rate if we are not able to control it. Being able to know
the pattern and trend of dengue cases will be of great significant in reducing the death rate
that will cause by dengue. A good and reliable mathematical modeling about pattern and
trend dengue will help the government to take preventing control and precaution control
to reduce the dengue case in certain time in the future since this disease does not have
specific treatment.

The objectives of this study are to know the pattern of dengue cases that happened in
chosen district, to obtain the first harmonic equation of Fourier series and compare the
peak value for dengue cases in the chosen district by referring to Fourier series graph
plotted. This project focuses on certain districts which had recorded the highest dengue
cases among district in Malaysia which are Shah Alam, Gombak and Klang.

3

2. METHODOLOGY

Some of Fourier series formula from Zill and Cullen (2009) that are used throughout this
study is given as follows:-

The Fourier series of a function f defined on the interval [0, 2L] is given by:
a 
 n n 
f ( x)  0    a n cos x  bn sin x
2 n1  L L 

where,
2L
1
a0 
L  f ( x)dx
0

n
2L
1
an 
L  f ( x) cos
0
L
xdx

n
2L
1
bn 
L  f ( x) sin
0
L
xdx

Fourier series determined from the coefficient which are a0, an, and bn. Since, we are
focus on the first harmonic term, we can write these coefficients as follow:
2L
1
a 0   f ( x)dx
L 0
1 
  yk
L k 1


y k
 k 1

L
 [average of f ( x)]

n
2L
1
a1 
L  f ( x) cos
0
L
xdx

1   nx  
   y k  cos
  

L  k 1   L  
   nx  
  y k  cos
  

 k 1   L  

L

4

n
2L
1
b1 
L  f ( x) sin
0
L
xdx

1   nx  
   y k  sin 
  

L  k 1   L  
   nx  
  y k  sin 
  

 k 1   L  

L

where yk is data obtained from the dengue cases and 2L is the period time. Then, we
arrange the Fourier series as follow:
a  x x   2x 2x 
f ( x)  0   a1 cos  b1 sin    a2 cos  b2 sin   ...
2  L L  L L 

 x x 
The term of  a1 cos  b1 sin  is called the first harmonic. We can write the sum of
 L L
sine and cosine term, with the same periodic as follow:
 x   2x 
y  f ( x)  c0  c1 sin  1   c2 sin   2   ...
L   L 
where,
a
c0  0 ,
2
c1  a12  b12 ,
 a1 
 1  tan 1 
 

 b1 

In this study, we focus on the first harmonic term on this equation which is:
 x 
y  c0  c1 sin
 L  1 

 

This equation is plotted by using Maple software to determine the peak value and analyze
the trend of dengue cases.

5

3. IMPLEMENTATION

Before proceed to Fourier series method, the data of dengue case were plotted by using
Excel in order to look for the pattern of dengue cases which happened in Shah Alam,
Gombak and Klang.

Figure 1. Graph of dengue cases in Shah Alam, Gombak and Klang for year 2009

Figure1 shows that comparison of dengue cases between Shah Alam, Gombak and Klang
since 7 January until 26 December 2009 by graph. From the graph above, it is hard to
compare the pattern between these districts. Thus, it is not accurate if we want to
generate the prediction for the next year based on the trend line equation. Furthermore,
there are a lot of scatter plot dengue cases data that fluctuations over the period cover.

6

Figure 2. Graph of dengue cases in Shah Alam, Gombak and Klang for year 2010

Figure2 shows that comparison of dengue cases between Shah Alam, Gombak and Klang
since 9 January until 8 August 2010 by graph. From the graph above, it is hard to
compare the pattern between these districts. Thus, it is not accurate if we want to
generate the prediction for the next year based on the trend line equation. Furthermore,
there are a lot of scatter plot dengue cases data that fluctuations over the period cover.

So, more suitable method to compare the pattern of number of dengue cases recorded in
Shah Alam, Gombak and Klang is Fourier series.

7

Table 1. Fourier series calculation by using excel for dengue cases in Shah Alam (2009)
Week (x) Cases (y) (πx)/L cos (πx/L) sin (πx/L) [cos ((πx)/L)] *yk [sin ((πx)/L)] *yk
1 244 0.1232 0.9924 0.1229 242.1506 29.9847
2 425 0.2464 0.9698 0.2439 412.1637 103.6633
3 362 0.3696 0.9325 0.3612 337.5549 130.7695
4 360 0.4928 0.8810 0.4731 317.1644 170.3137
5 474 0.6160 0.8162 0.5778 386.8773 273.8648
6 337 0.7392 0.7390 0.6737 249.0460 227.0354
7 525 0.8624 0.6506 0.7594 341.5746 398.6876
8 482 0.9856 0.5524 0.8336 266.2399 401.7963
9 489 1.1088 0.4457 0.8952 217.9661 437.7348
10 608 1.2320 0.3324 0.9432 202.0717 573.4379
11 521 1.3552 0.2139 0.9768 111.4591 508.9380
12 622 1.4784 0.0923 0.9957 57.3909 619.3467
13 552 1.6016 -0.0308 0.9995 -16.9989 551.7382
14 383 1.7248 -0.1534 0.9882 -58.7490 378.4674
15 260 1.8480 -0.2737 0.9618 -71.1524 250.0747
16 360 1.9712 -0.3898 0.9209 -140.3229 331.5260
17 41 2.0944 -0.5000 0.8660 -20.5000 35.5070
18 132 2.2176 -0.6026 0.7980 -79.5478 105.3383
19 127 2.3408 -0.6961 0.7179 -88.4090 91.1748
20 76 2.4640 -0.7791 0.6269 -59.2101 47.6462
21 102 2.5872 -0.8502 0.5264 -86.7221 53.6961
22 79 2.7104 -0.9085 0.4180 -71.7688 33.0189
23 103 2.8336 -0.9529 0.3032 -98.1530 31.2247
24 65 2.9568 -0.9830 0.1837 -63.8933 11.9437
25 51 3.0800 -0.9981 0.0616 -50.9033 3.1396
26 76 3.2032 -0.9981 -0.0616 -75.8559 -4.6786
27 9 3.3264 -0.9830 -0.1837 -8.8468 -1.6537
28 46 3.4496 -0.9529 -0.3032 -43.8353 -13.9450
29 27 3.5728 -0.9085 -0.4180 -24.5286 -11.2849
30 16 3.6960 -0.8502 -0.5264 -13.6035 -8.4229
31 11 3.8192 -0.7791 -0.6269 -8.5699 -6.8962
32 11 3.9424 -0.6961 -0.7179 -7.6575 -7.8970
33 0 4.0656 -0.6026 -0.7980 0.0000 0.0000
34 11 4.1888 -0.5000 -0.8660 -5.5000 -9.5263
35 0 4.3120 -0.3898 -0.9209 0.0000 0.0000
36 0 4.4352 -0.2737 -0.9618 0.0000 0.0000
37 42 4.5584 -0.1534 -0.9882 -6.4424 -41.5029
38 56 4.6816 -0.0308 -0.9995 -1.7245 -55.9734
39 32 4.8048 0.0923 -0.9957 2.9526 -31.8635
40 48 4.9280 0.2139 -0.9768 10.2688 -46.8887
41 40 5.0512 0.3324 -0.9432 13.2942 -37.7262
42 0 5.1744 0.4457 -0.8952 0.0000 0.0000
43 0 5.2976 0.5524 -0.8336 0.0000 0.0000
44 0 5.4208 0.6506 -0.7594 0.0000 0.0000
45 0 5.5440 0.7390 -0.6737 0.0000 0.0000
46 0 5.6672 0.8162 -0.5778 0.0000 0.0000
47 0 5.7904 0.8810 -0.4731 0.0000 0.0000
48 0 5.9136 0.9325 -0.3612 0.0000 0.0000
49 0 6.0368 0.9698 -0.2439 0.0000 0.0000
50 0 6.1600 0.9924 -0.1229 0.0000 0.0000
51 0 6.2832 1.0000 0.0000 0.0000 0.0000
TOTAL 8205 0.0000 0.0000 2065.2801 5521.8088

8

Table 1 shows that the calculations for Fourier series by using Excel. From the table, we
can obtain coefficients of Fourier series which are a0, a1 and b1 where period time, L is
25.5 which is half of 51 (numbers of data).

1 51
a0   yk
L k 1
8205

51
 160.8824

 51   nx  
  y k  cos
  

 k 1   L  
a1 
L
2605.2801

25.5
 80.9914

 51   nx  
  y k  sin 
  

 k 1   L  
b1 
L
5521.808

25.5
 216.5415

Then, we arrange the Fourier series as follow:

160.8824  x x 
f ( x)    80.9914 cos  216.5415 sin   ...
2  25.5 25.5 

9

We can write the sum of sine and cosine term, with the same periodic which focus on the
first harmonic by calculated the value of c0, c1 and α1.

a0
c0 
2
160.8824

2
 80.4412

c1  a12  b12 ,
 80.9914 2  216.5415 2
 231.1922

 a1 
 1  tan 1 
 

 b1 
 80.9914 
 tan 1  
 216.5415 
 0.3579
Then,
 x 
y  80.4412  231.1922 sin 
 L  0.3579 
 


10

>
>

>

Figure 3. Fourier series graph plotted for dengue cases in Shah Alam (2009)

Figure 3 shows the Fourier series plotted with Maple software in first harmonic. The y-
axis represents the number of dengue cases and the x-axis represents the number of
weeks. From the graph, it shows that the maximum point or peak point in week 10 with
310 dengue cases. However, between week 26 to week 45, the graph shows that the
minimum number of case which is zero. It happened because the different or gap
between actual data for maximum cases and minimum cases is high. Early hypothesis
from this graph is the highest cases happen in week 10 with 310 cases and the lowest
cases happen between week 26 to week 45.

11

Table 2. Fourier series calculation by using excel for dengue cases in Gombak (2009)
Week (x) Cases (y) (πx)/L cos [(πx)/L] sin [(πx)/L ] [cos [(πx)/L]]*yk [[sin (πx/L)] *yk]
1 257 0.1232 0.9924 0.1229 255.0521 31.5823
2 217 0.2464 0.9698 0.2439 210.4459 52.9293
3 158 0.3696 0.9325 0.3612 147.3306 57.0762
4 118 0.4928 0.8810 0.4731 103.9594 55.8250
5 136 0.6160 0.8162 0.5778 111.0028 78.5772
6 142 0.7392 0.7390 0.6737 104.9393 95.6648
7 190 0.8624 0.6506 0.7594 123.6175 144.2869
8 80 0.9856 0.5524 0.8336 44.1892 66.6882
9 274 1.1088 0.4457 0.8952 122.1323 245.2747
10 269 1.2320 0.3324 0.9432 89.4034 253.7085
11 362 1.3552 0.2139 0.9768 77.4438 353.6191
12 364 1.4784 0.0923 0.9957 33.5857 362.4472
13 381 1.6016 -0.0308 0.9995 -11.7329 380.8193
14 267 1.7248 -0.1534 0.9882 -40.9556 263.8402
15 256 1.8480 -0.2737 0.9618 -70.0577 246.2274
16 240 1.9712 -0.3898 0.9209 -93.5486 221.0173
17 19 2.0944 -0.5000 0.8660 -9.5000 16.4545
18 160 2.2176 -0.6026 0.7980 -96.4215 127.6828
19 17 2.3408 -0.6961 0.7179 -11.8343 12.2045
20 17 2.4640 -0.7791 0.6269 -13.2444 10.6577
21 17 2.5872 -0.8502 0.5264 -14.4537 8.9493
22 58 2.7104 -0.9085 0.4180 -52.6910 24.2417
23 57 2.8336 -0.9529 0.3032 -54.3177 17.2797
24 63 2.9568 -0.9830 0.1837 -61.9273 11.5762
25 67 3.0800 -0.9981 0.0616 -66.8729 4.1246
26 79 3.2032 -0.9981 -0.0616 -78.8502 -4.8633
27 85 3.3264 -0.9830 -0.1837 -83.5527 -15.6187
28 100 3.4496 -0.9529 -0.3032 -95.2942 -30.3153
29 189 3.5728 -0.9085 -0.4180 -171.6999 -78.9945
30 131 3.6960 -0.8502 -0.5264 -111.3784 -68.9626
31 111 3.8192 -0.7791 -0.6269 -86.4779 -69.5885
32 104 3.9424 -0.6961 -0.7179 -72.3979 -74.6628
33 61 4.0656 -0.6026 -0.7980 -36.7607 -48.6791
34 91 4.1888 -0.5000 -0.8660 -45.5000 -78.8083
35 59 4.3120 -0.3898 -0.9209 -22.9974 -54.3334
36 69 4.4352 -0.2737 -0.9618 -18.8827 -66.3660
37 90 4.5584 -0.1534 -0.9882 -13.8052 -88.9349
38 81 4.6816 -0.0308 -0.9995 -2.4944 -80.9616
39 59 4.8048 0.0923 -0.9957 5.4438 -58.7483
40 63 4.9280 0.2139 -0.9768 13.4778 -61.5414
41 113 5.0512 0.3324 -0.9432 37.5561 -106.5765
42 67 5.1744 0.4457 -0.8952 29.8645 -59.9759
43 109 5.2976 0.5524 -0.8336 60.2078 -90.8627
44 51 5.4208 0.6506 -0.7594 33.1815 -38.7297
45 42 5.5440 0.7390 -0.6737 31.0384 -28.2952
46 65 5.6672 0.8162 -0.5778 53.0528 -37.5553
47 53 5.7904 0.8810 -0.4731 46.6936 -25.0740
48 50 5.9136 0.9325 -0.3612 46.6236 -18.0621
49 20 6.0368 0.9698 -0.2439 19.3959 -4.8783
50 22 6.1600 0.9924 -0.1229 21.8333 -2.7035
51 14 6.2832 1.0000 0.0000 14.0000 0.0000
TOTAL 6164 0.0000 0.0000 397.8217 1848.6629

12

Table 2 shows that the calculations for Fourier series by using excel. From the table, we

1 51
a0   yk
L k 1
6164

51
 120.8627

 51   nx  
  y k  cos
  

 k 1   L  
a1 
L
397.8217

25.5
 15.6009

 51   nx  
  y k  sin 
  

 k 1   L  
b1 
L
1848.6629

25.5
 72.4966


120.8627  x x 
f ( x)   15.6009 cos  72.4966 sin   ...
2  25.5 25.5 

13


a0
c0 
2
120.8627

2
 60.4314

c1  a12  b12 ,
 15.6009 2  72.4966 2
 74.1562

 a1 
 1  tan 1 
 

 b1 
 15.6009 
 tan 1  
 72.4966 
 0.2120
Then,
 x 
y  60.4314  74.1562 sin 
 L  0.2120 
 


14

>
>

>

Figure 4. Fourier series graph plotted for dengue cases in Gombak (2009)

Figure 4 shows that Fourier series plotted with Maple software in first harmonic. The y-
axis represents the number of dengue cases and the x-axis represents the number of
134 dengue cases. However, from week 32 to week 42, the graph shows that the
cases happen between week 32 to week 42.

15

Table 3. Fourier series calculations by using excel for dengue cases in Klang (2009)
Week(x) Cases (y) (πx)/L cos (πx/L) sin (πx/L) [cos ((πx)/L )]*yk [sin ((πx)/L)] *yk
1 0 0.1232 0.9924 0.1229 0.0000 0.0000
2 116 0.2464 0.9698 0.2439 112.4964 28.2940
3 41 0.3696 0.9325 0.3612 38.2314 14.8109
4 64 0.4928 0.8810 0.4731 56.3848 30.2780
5 0 0.6160 0.8162 0.5778 0.0000 0.0000
6 22 0.7392 0.7390 0.6737 16.2582 14.8213
7 37 0.8624 0.6506 0.7594 24.0729 28.0980
8 49 0.9856 0.5524 0.8336 27.0659 40.8465
9 84 1.1088 0.4457 0.8952 37.4420 75.1937
10 78 1.2320 0.3324 0.9432 25.9237 73.5660
11 145 1.3552 0.2139 0.9768 31.0203 141.6430
12 189 1.4784 0.0923 0.9957 17.4387 188.1938
13 178 1.6016 -0.0308 0.9995 -5.4815 177.9156
14 210 1.7248 -0.1534 0.9882 -32.2122 207.5147
15 236 1.8480 -0.2737 0.9618 -64.5845 226.9909
16 334 1.9712 -0.3898 0.9209 -130.1885 307.5824
17 125 2.0944 -0.5000 0.8660 -62.5000 108.2532
18 178 2.2176 -0.6026 0.7980 -107.2690 142.0471
19 130 2.3408 -0.6961 0.7179 -90.4974 93.3285
20 109 2.4640 -0.7791 0.6269 -84.9198 68.3347
21 89 2.5872 -0.8502 0.5264 -75.6693 46.8525
22 60 2.7104 -0.9085 0.4180 -54.5079 25.0776
23 48 2.8336 -0.9529 0.3032 -45.7412 14.5513
24 20 2.9568 -0.9830 0.1837 -19.6595 3.6750
25 9 3.0800 -0.9981 0.0616 -8.9829 0.5540
26 9 3.2032 -0.9981 -0.0616 -8.9829 -0.5540
27 4 3.3264 -0.9830 -0.1837 -3.9319 -0.7350
28 6 3.4496 -0.9529 -0.3032 -5.7177 -1.8189
29 6 3.5728 -0.9085 -0.4180 -5.4508 -2.5078
30 0 3.6960 -0.8502 -0.5264 0.0000 0.0000
31 0 3.8192 -0.7791 -0.6269 0.0000 0.0000
32 0 3.9424 -0.6961 -0.7179 0.0000 0.0000
33 2 4.0656 -0.6026 -0.7980 -1.2053 -1.5960
34 0 4.1888 -0.5000 -0.8660 0.0000 0.0000
35 0 4.3120 -0.3898 -0.9209 0.0000 0.0000
36 0 4.4352 -0.2737 -0.9618 0.0000 0.0000
37 8 4.5584 -0.1534 -0.9882 -1.2271 -7.9053
38 0 4.6816 -0.0308 -0.9995 0.0000 0.0000
39 0 4.8048 0.0923 -0.9957 0.0000 0.0000
40 0 4.9280 0.2139 -0.9768 0.0000 0.0000
41 0 5.0512 0.3324 -0.9432 0.0000 0.0000
42 0 5.1744 0.4457 -0.8952 0.0000 0.0000
43 0 5.2976 0.5524 -0.8336 0.0000 0.0000
44 0 5.4208 0.6506 -0.7594 0.0000 0.0000
45 0 5.5440 0.7390 -0.6737 0.0000 0.0000
46 0 5.6672 0.8162 -0.5778 0.0000 0.0000
47 0 5.7904 0.8810 -0.4731 0.0000 0.0000
48 0 5.9136 0.9325 -0.3612 0.0000 0.0000
49 16 6.0368 0.9698 -0.2439 15.5168 -3.9026
50 8 6.1600 0.9924 -0.1229 7.9394 -0.9831
51 0 6.2832 1.0000 0.0000 0.0000 0.0000
TOTAL 2610 0.0000 0.0000 -398.9390 2038.4200

16


1 51
a0   yk
L k 1
2610

51
 51.1765

 51   nx  
  y k  cos
  

 k 1   L  
a1 
L
 398.9390

25.5
 15.6447

 51   nx  
  y k  sin 
  

 k 1   L  
b1 
L
2038.4200

25.5
 79.9380


51.1765  x x 
f ( x)     15.6447 cos  79.9380 sin   ...
2  25.5 25.5 

17


a0
c0 
2
51.1765

2
 25.5882

c1  a12  b12 ,
 (15.6447) 2  (79.9380) 2
 81.4546

 a1 
 1  tan 1 
 

 b1 
  15.6447 
 tan 1  
 79.9380 
 0.1933
Then,
 x 
y  25.5882  81.4546 sin
 L  0.1933 

 


18

>
>

>

Figure 5. Fourier series graph plotted for dengue cases in Klang (2009)

Figure 5 shows that Fourier series that plotted with Maple software in first harmonic. For
y-axis represents the number of dengue cases and for x-axis represents the number of
cases happen between weeks 30 to week 50.

19

Table 4. Fourier series calculations for dengue cases in Shah Alam (2010)

Week (x) Total (y) (πx)/L cos [(πx)/L] sin [(πx)/L] cos [(πx)/L] *Yk sin [(πx)/L] *Yk
1 56 0.2027 0.9795 0.2013 54.8537 11.2727
2 77 0.4054 0.9190 0.3944 70.7598 30.3654
3 85 0.6081 0.8208 0.5713 69.7649 48.5578
4 140 0.8107 0.6890 0.7248 96.4554 101.4710
5 168 1.0134 0.5290 0.8486 88.8660 142.5722
6 172 1.2161 0.3473 0.9378 59.7365 161.2934
7 168 1.4188 0.1514 0.9885 25.4399 166.0627
8 166 1.6215 -0.0506 0.9987 -8.4078 165.7869
9 162 1.8242 -0.2507 0.9681 -40.6057 156.8285
10 119 2.0268 -0.4404 0.8978 -52.4069 106.8387
11 109 2.2295 -0.6121 0.7908 -66.7196 86.1946
12 110 2.4322 -0.7588 0.6514 -83.4634 71.6510
13 61 2.6349 -0.8743 0.4853 -53.3351 29.6034
14 82 2.8376 -0.9541 0.2994 -78.2394 24.5478
15 74 3.0403 -0.9949 0.1012 -73.6203 7.4865
16 69 3.2429 -0.9949 -0.1012 -68.6460 -6.9806
17 44 3.4456 -0.9541 -0.2994 -41.9821 -13.1720
18 41 3.6483 -0.8743 -0.4853 -35.8482 -19.8974
19 14 3.8510 -0.7588 -0.6514 -10.6226 -9.1192
20 14 4.0537 -0.6121 -0.7908 -8.5695 -11.0709
21 10 4.2564 -0.4404 -0.8978 -4.4039 -8.9780
22 13 4.4590 -0.2507 -0.9681 -3.2585 -12.5850
23 5 4.6617 -0.0506 -0.9987 -0.2532 -4.9936
24 6 4.8644 0.1514 -0.9885 0.9086 -5.9308
25 22 5.0671 0.3473 -0.9378 7.6407 -20.6305
26 29 5.2698 0.5290 -0.8486 15.3400 -24.6107
27 34 5.4725 0.6890 -0.7248 23.4249 -24.6430
28 27 5.6751 0.8208 -0.5713 22.1606 -15.4242
29 21 5.8778 0.9190 -0.3944 19.2981 -8.2815
30 29 6.0805 0.9795 -0.2013 28.4064 -5.8377
31 23 6.2832 1.0000 0.0000 23.0000 0.0000
TOTAL 2150 0.0000 0.0000 -24.3271 1118.3775

20


1 31
a0   yk
L k 1
2150

31
 69.3548

 31   nx  
  y k  cos
  

 k 1   L  
a1 
L
 24.3271

15.5
 1.5695

 31   nx  
  y k  sin 
  

 k 1   L  
b1 
L
1118.3775

15.5
 72.1534


69.3548  x x 
f ( x)     1.5695 cos  72.1534 sin   ...
2  15.5 15.5 

21


a0
c0 
2
69.3548

2
 34.6774

c1  a12  b12 ,
 (1.5695) 2  (72.1534) 2
 72.1705

 a1 
 1  tan 1 
 

 b1 
  1.5695 
 tan 1  
 72.1534 
 0.0217
Then,
 x 
y  34.6774  72.1705 sin
 L  0.0217 

 


22

>
>

>

Figure 6. Fourier series graph plotted for dengue cases in Shah Alam (2010)

Figure 6 shows that Fourier series that plotted with Maple software in first harmonic. The
y-axis represents the number of dengue cases and the x-axis represents the number of
from this graph is the highest cases happen in week 9 with 120 cases and the lowest cases
happen between week 18 to week 28.

23

Table 5. Fourier series calculations by using excel for dengue cases in Gombak (2010)
Week (x) Cases (y) (πx)/L cos [(πx)/L] sin [(πx)/L] [cos [(πx)/L]]*yk [sin [(πx)/L]] *yk
1 118 0.2027 0.9795 0.2013 115.5845 23.7532
2 126 0.4054 0.9190 0.3944 115.7887 49.6888
3 117 0.6081 0.8208 0.5713 96.0293 66.8384
4 184 0.8107 0.6890 0.7248 126.7699 133.3619
5 185 1.0134 0.5290 0.8486 97.8583 156.9992
6 184 1.2161 0.3473 0.9378 63.9042 172.5464
7 183 1.4188 0.1514 0.9885 27.7113 180.8897
8 174 1.6215 -0.0506 0.9987 -8.8130 173.7767
9 192 1.8242 -0.2507 0.9681 -48.1253 185.8708
10 180 2.0268 -0.4404 0.8978 -79.2709 161.6048
11 188 2.2295 -0.6121 0.7908 -115.0759 148.6658
12 166 2.4322 -0.7588 0.6514 -125.9538 108.1278
13 100 2.6349 -0.8743 0.4853 -87.4347 48.5302
14 125 2.8376 -0.9541 0.2994 -119.2674 37.4204
15 77 3.0403 -0.9949 0.1012 -76.6049 7.7900
16 92 3.2429 -0.9949 -0.1012 -91.5280 -9.3075
17 74 3.4456 -0.9541 -0.2994 -70.6063 -22.1529
18 67 3.6483 -0.8743 -0.4853 -58.5812 -32.5152
19 58 3.8510 -0.7588 -0.6514 -44.0080 -37.7796
20 52 4.0537 -0.6121 -0.7908 -31.8295 -41.1203
21 46 4.2564 -0.4404 -0.8978 -20.2581 -41.2990
22 46 4.4590 -0.2507 -0.9681 -11.5300 -44.5315
23 54 4.6617 -0.0506 -0.9987 -2.7351 -53.9307
24 55 4.8644 0.1514 -0.9885 8.3285 -54.3658
25 66 5.0671 0.3473 -0.9378 22.9221 -61.8916
26 67 5.2698 0.5290 -0.8486 35.4406 -56.8592
27 86 5.4725 0.6890 -0.7248 59.2512 -62.3322
28 109 5.6751 0.8208 -0.5713 89.4632 -62.2682
29 130 5.8778 0.9190 -0.3944 119.4645 -51.2663
30 138 6.0805 0.9795 -0.2013 135.1751 -27.7792
31 132 6.2832 1.0000 0.0000 132.0000 0.0000
TOTAL 3571 0.0000 0.0000 254.0694 996.4649

24


1 31
a0   yk
L k 1
3571

31
 115.1935

 31   nx  
  y k  cos
  

 k 1   L  
a1 
L
254.0694

15.5
 16.3916

 31   nx  
  y k  sin 
  

 k 1   L  
b1 
L
996.4649

15.5
 64.2881


115.1935  x x 
f ( x)   16.3981cos  64.2881sin   ...
2  15.5 15.5 

25


a0
c0 
2
115.1935

2
 57.5968

c1  a12  b12 ,
 (16.3916) 2  (64.2881) 2
 66.3448

 a1 
 1  tan 1 
 

 b1 
 16.3916 
 tan 1  
 64.2881 
 0.2497
Then,
 x 
y  57.5968  66.3448 sin
 L  0.2497 

 


26

>
>

>

Figure 7. Fourier series graph plotted for dengue cases in Gombak (2010)

from this graph is the highest cases happen in week 7 with 120 cases and the lowest cases
happen between week 19 to week 24.

27

Table 6. Fourier series calculations by using excel for dengue cases in Klang (2010)

Week(x) Total (y) (πx)/L cos (πx)/L sin (πx)/L [cos [(πx)/L]] *yk [sin [(πx)/L]] *yk
1 20 0.2027 0.9795 0.2013 19.5906 4.0260
2 14 0.4054 0.9190 0.3944 12.8654 5.5210
3 17 0.6081 0.8208 0.5713 13.9530 9.7116
4 40 0.8107 0.6890 0.7248 27.5587 28.9917
5 48 1.0134 0.5290 0.8486 25.3903 40.7349
6 47 1.2161 0.3473 0.9378 16.3233 44.0744
7 53 1.4188 0.1514 0.9885 8.0257 52.3888
8 59 1.6215 -0.0506 0.9987 -2.9883 58.9243
9 43 1.8242 -0.2507 0.9681 -10.7781 41.6273
10 53 2.0268 -0.4404 0.8978 -23.3409 47.5836
11 51 2.2295 -0.6121 0.7908 -31.2174 40.3296
12 47 2.4322 -0.7588 0.6514 -35.6616 30.6145
13 38 2.6349 -0.8743 0.4853 -33.2252 18.4415
14 41 2.8376 -0.9541 0.2994 -39.1197 12.2739
15 50 3.0403 -0.9949 0.1012 -49.7435 5.0584
16 53 3.2429 -0.9949 -0.1012 -52.7281 -5.3619
17 20 3.4456 -0.9541 -0.2994 -19.0828 -5.9873
18 25 3.6483 -0.8743 -0.4853 -21.8587 -12.1325
19 29 3.8510 -0.7588 -0.6514 -22.0040 -18.8898
20 24 4.0537 -0.6121 -0.7908 -14.6905 -18.9786
21 21 4.2564 -0.4404 -0.8978 -9.2483 -18.8539
22 12 4.4590 -0.2507 -0.9681 -3.0078 -11.6169
23 13 4.6617 -0.0506 -0.9987 -0.6584 -12.9833
24 14 4.8644 0.1514 -0.9885 2.1200 -13.8386
25 15 5.0671 0.3473 -0.9378 5.2096 -14.0663
26 19 5.2698 0.5290 -0.8486 10.0503 -16.1242
27 15 5.4725 0.6890 -0.7248 10.3345 -10.8719
28 16 5.6751 0.8208 -0.5713 13.1322 -9.1403
29 13 5.8778 0.9190 -0.3944 11.9465 -5.1266
30 31 6.0805 0.9795 -0.2013 30.3654 -6.2403
31 33 6.2832 1.0000 0.0000 33.0000 0.0000
TOTAL 974 0.0000 0.0000 -129.4878 260.0890

28


1 31
a0   yk
L k 1
974

31
 31.4194

 31   nx  
  y k  cos
  

 k 1   L  
a1 
L
 129.4878

15.5
 8.3541

 31   nx  
  y k  sin 
  

 k 1   L  
b1 
L
260.0890

15.5
 16.7799


31.4194  x x 
f ( x)     8.3541cos  16.7799 sin   ...
2  15.5 15.5 

29


a0
c0 
2
31.4194

2
 15.7097

c1  a12  b12 ,
 (8.3541) 2  (16.7799) 2
 18.7445

 a1 
 1  tan 1 
 

 b1 
  8.3541 
 tan 1  
 16.7799 
 0.4619
Then,
 x 
y  15.7097  18.7445 sin
 L  0.4619 

 


30

>
>

>

Figure 8. Fourier series graph plotted for dengue cases in Klang (2010)

35 dengue cases. However, from week 23 to week 28, the graph shows that the minimum
number of case which is zero. It happened because the different or gap between actual
data for maximum cases and minimum cases is high. Early hypothesis from this graph is
the highest cases happen in week 10 with 35 cases and the lowest cases happen between
week 23 to week 28.

31

4. RESULTS AND DISCUSSION

From the findings, it can be noticed that the data of dengue cases in these three districts
which are Shah Alam, Gombak and Klang is distributed fluctuation. It is difficult to
determine and predict the dengue cases for the next year by following the trend line that is
generated by Excel.

Table 7. Fourier series equations on 1st harmonic for 2009
District Fourier Series Equation

Shah Alam y= 80.4412 + 231.1922sin[(πx/25.5) + 0.3579]
Gombak y= 60.4314 + 74.1562 sin[(πx/25.5) + 0.2120]
Klang y= 25.5882 + 81.4546 sin[(πx/25.5) -0.1933]

Table 8. Fourier series equations on 1st harmonic for 2010
District Fourier Series Equation

Shah Alam y= 34.6774 + 72.1705sin [(πx/15.5) – 0.0217]
Gombak y= 57.5968 + 66.344sin [(πx/15.5) +0.2497]
Klang y= 15.7097 + 18.7445sin [(πx/15.5) – 0.4619]

Table 9. Analysis from graph using maple software for 2009
District Peak Value Cases Week

Shah Alam 310 10
Gombak 130 10
Klang 110 12

Table 10. Analysis from graph using maple software for 2010
District Peak Value Cases Week

Shah Alam 120 9
Gombak 120 7
Klang 35 10

32

The equation of Fourier series on first harmonic for dengue cases are shown in Table 7
and Table 8. For the year 2009, dengue cases seasonally peak between period week 10 to
14 (14 March 2009 until 11 April 2009) averagely recorded between 100 and 300 cases
per week. It also shows that dengue cases dengue cases slowly decrease for chosen
district at the end of year 2009. For the year 2010, dengue cases seasonally peak between
period week 7 to week 10 (20 February 2010 until 13 March 2010) averagely recorded
between 35 and 120 cases per week. It also shows that dengue cases dengue cases slowly
decrease for chosen district started from week 18 to week 28 (8 May 2010 to 18 July).

If we want to compare the result between year 2009 and 2010, we can see dengue cases
seasonally peak at first quarter of year which averagely recorded in period week 7 to
week 14 (February to April). Then, the dengue cases will reduce slowly in the third
quarter of the year.

5. CONCLUSIONS AND RECOMMENDATIONS

Dengue is one of the diseases with no specific treatment or immunizations. Thus, the
preventive precautions from dengue such as fogging are important to reduce the cases.

We can summarize that the peak dengue cases is peak between first quarter of the year
which averagely recorded in period week 7 to week 14 (February to April). Shah Alam
recorded the highest dengue cases in 2009 which 310 cases in week 10 compared to
Klang which recorded 110. In year 2010, Shah Alam and Klang show drastic decrease the
number of cases which 120 and 35 cases respectively. However, Gombak did not record
the decrease cases in year 2010 compared to year 2009 which gives average of 120 cases.

From the findings, it is recommended that the Ministry of Health Malaysia should focus
more on first quarter of the year (February until April) every year to reduce dengue cases
because this period recorded highest cases in 2009 and 2010.

Further studies can be done for the previous year such as 2008 or 2007. So, the seasonal
peak can be determined further. This model can be explored further by comparing the
dengue cases recorded with climatic variability that is rainfalls, temperature and vapor
pressure in those selected districts in Selangor. Comparison can also be done between
states in Malaysia.

33

REFERENCES

Abas N., Daud Z.M., Yusof F. (2009). Fourier Series In A Temporal Rainfall Model.
Proceeding of the 5th Asian Mathematical Conference, Malaysia.

Ang, K.C. & Li, Zi. (1999). Modeling The Spread of Dengue in Singapore, Division of
Mathematics, School of Sciences, Nanyang Technological University, Singapore.

Angove, C. (2009). Some Discrete Real And Complex Fourier Transforms, A Discussion,
With Examples.

Favier, C., Degallier, N. & Dubois, M.A. (2005). Dengue Epidemic Modelling: Stakes
and Pitfalls.

Klingenberg, L. (2005). Frequency Domain Using Excel. San Francisco State University
School of Engineering.

Kvernadze, G., Hagstrom, T., and Shapiro, H. (2000). Detecting the Singularities of a
Function of VP Class by its Integrated Fourier Series. Computers and Mathematics with
Applications, 39, 25-43.

McNeal, J. D. and Zeytuncu, Y. U. (2006). A note on rearrangement of Fourier
series. J. Math. Anal. Appl. 323 (2006) 1348–1353.

Nuraini, N., Soewono, E. & Sidarto, K.A. (2006). Mathematical Model of Dengue
Disease Transmission with Severe DHF Compartment, Bulletin of the Malaysian
Mathematical Sciences Society, 30, 143-157.

Pongsumpun, P. & Tang, I.M. (2001). A Realistic Age Structure Transmission Model for
Dengue Hemorrhagic Fever in Thailand, Department of Mathematics and Physics,
Faculty of Sciences, Mahidol University.

Tamrin, H., Riyanto, M.Z., Akhid Ardhian, A. (2007). Not Fatal Disease For SIR Model.

Zill, D.G. & Cullen, M. R. (2009). Differential Equations With Boundary-Value
Problems, 7th Edition, International Student Edition.

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