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.

ii

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

iii

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

iv

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).

1

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.

2

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.

3

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.

4

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:

1

0 sincos2

)(n

nn xL

nbx

L

na

axf

where,

L

dxxfL

a

2

0

0 )(1

xdxL

nxf

La

L

n

2

0

cos)(1

xdxL

nxf

Lb

L

n

2

0

sin)(1

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:

)]([

L

y

1

)(1

1k

k

1

2

0

0

xfofaverage

yL

dxxfL

a

k

k

L

L

L

xny

L

xny

L

xdxL

nxf

La

k

k

k

k

L

1

1

2

0

1

cos

cos1

cos)(1

5

L

L

xny

L

xny

L

xdxL

nxf

Lb

k

k

k

k

L

1

1

2

0

1

sin

sin1

sin)(1

where yk is data obtained from the dengue cases and 2L is the period time. Then, we

arrange the Fourier series as follow:

...2

sin2

cossincos2

)( 2211

0

L

xb

L

xa

L

xb

L

xa

axf

The term of

L

xb

L

xa

sincos 11 is called the first harmonic. We can write the sum of

sine and cosine term, with the same periodic as follow:

...2

sinsin)( 22110

L

xc

L

xccxfy

where,

1

11

1

2

1

2

11

0

0

tan

,

,2

b

a

bac

ac

In this study, we focus on the first harmonic term on this equation which is:

110 sin

L

xccy

This equation is plotted by using Maple software to determine the peak value and analyze

the trend of dengue cases.

6

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.

7

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.

8

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

9

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).

8824.160

51

8205

1 51

1

0

k

kyL

a

9914.80

5.25

2801.2605

cos51

1

1

L

L

xny

ak

k

5415.216

5.25

808.5521

sin51

1

1

L

L

xny

bk

k

Then, we arrange the Fourier series as follow:

...5.25

sin5415.2165.25

cos9914.802

8824.160)(

xxxf

10

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.

4412.80

2

8824.160

2

0

0

a

c

1922.231

5415.2169914.80

,

22

2

1

2

11

bac

3579.0

5415.216

9914.80tan

tan

1

1

11

1

b

a

Then,

3579.0sin1922.2314412.80

L

xy



11

>

>

>

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.

12

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

13

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



8627.120

51

6164

1 51

1

0

k

kyL

a

6009.15

5.25

8217.397

cos51

1

1

L

L

xny

ak

k

4966.72

5.25

6629.1848

sin51

1

1

L

L

xny

bk

k


...5.25

sin4966.725.25

cos6009.152

8627.120)(

xxxf

14



4314.60

2

8627.120

2

0

0

a

c

1562.74

4966.726009.15

,

22

2

1

2

11

bac

2120.0

4966.72

6009.15tan

tan

1

1

11

1

b

a

Then,

2120.0sin1562.744314.60

L

xy



15

>

>

>

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.

16

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

17




1765.51

51

2610

1 51

1

0

k

kyL

a

6447.15

5.25

9390.398

cos51

1

1

L

L

xny

ak

k

9380.79

5.25

4200.2038

sin51

1

1

L

L

xny

bk

k


...5.25

sin9380.795.25

cos6447.152

1765.51)(

xxxf

18



5882.25

2

1765.51

2

0

0

a

c

4546.81

)9380.79()6447.15(

,

22

2

1

2

11

bac

1933.0

9380.79

6447.15tan

tan

1

1

11

1

b

a

Then,

1933.0sin4546.815882.25

L

xy



19

>

>

>

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.

20

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

21




3548.69

31

2150

1 31

1

0

k

kyL

a

5695.1

5.15

3271.24

cos31

1

1

L

L

xny

ak

k

1534.72

5.15

3775.1118

sin31

1

1

L

L

xny

bk

k


...5.15

sin1534.725.15

cos5695.12

3548.69)(

xxxf

22



6774.34

2

3548.69

2

0

0

a

c

1705.72

)1534.72()5695.1(

,

22

2

1

2

11

bac

0217.0

1534.72

5695.1tan

tan

1

1

11

1

b

a

Then,

0217.0sin1705.726774.34

L

xy



23

>

>

>

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.

24

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

25




1935.115

31

3571

1 31

1

0

k

kyL

a

3916.16

5.15

0694.254

cos31

1

1

L

L

xny

ak

k

2881.64

5.15

4649.996

sin31

1

1

L

L

xny

bk

k


...5.15

sin2881.645.15

cos3981.162

1935.115)(

xxxf

26



5968.57

2

1935.115

2

0

0

a

c

3448.66

)2881.64()3916.16(

,

22

2

1

2

11

bac

2497.0

2881.64

3916.16tan

tan

1

1

11

1

b

a

Then,

2497.0sin3448.665968.57

L

xy



27

>

>

>

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.

28

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

29




4194.31

31

974

1 31

1

0

k

kyL

a

3541.8

5.15

4878.129

cos31

1

1

L

L

xny

ak

k

7799.16

5.15

0890.260

sin31

1

1

L

L

xny

bk

k


...5.15

sin7799.165.15

cos3541.82

4194.31)(

xxxf

30



7097.15

2

4194.31

2

0

0

a

c

7445.18

)7799.16()3541.8(

,

22

2

1

2

11

bac

4619.0

7799.16

3541.8tan

tan

1

1

11

1

b

a

Then,

4619.0sin7445.187097.15

L

xy



31

>

>

>

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.

32

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

33

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.

34

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.

Technology

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