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

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

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

  1. 1. ACKNOWLEDGEMENTSIN THE NAME OF ALLAH, THE MOST GRACIOUS, THE MOST MERCIFULFirstly, I am grateful to Allah S.W.T for giving me the strength to complete this projectsuccessfully. I would like to express my gratitude Associated Professor MaheranNuruddin for the guidance and support to this report.Special thanks to my family, especially my parent who always support and pray for mysuccess. I wish to thank my friends for their support. They have been very supportivethroughout the completion of this project.Last but not least, thank you again for those who spent time and effort with me incompleting this report, directly or indirectly. Without Allah bless and the all kindness ofthese people, I will never succeed in completing this project. Thank you so much.Wasalam.
  2. 2. TABLE OF CONTENTSACKNOWLEDGEMENTS ................................................................................................. iTABLE OF CONTENTS .................................................................................................... iiLIST OF TABLES ............................................................................................................. iiiLIST OF FIGURES ........................................................................................................... iiiABSTRACT ....................................................................................................................... iv1. INTRODUCTION ........................................................................................................ 12. METHODOLOGY ....................................................................................................... 43. IMPLEMENTATION ................................................................................................... 64. RESULTS AND DISCUSSION ................................................................................. 325. CONCLUSIONS AND RECOMMENDATIONS ..................................................... 33REFERENCES ................................................................................................................. 34 ii
  3. 3. LIST OF TABLESTable 1. Fourier series calculation by using excel for dengue cases in Shah Alam (2009) . 8Table 2. Fourier series calculation by using excel for dengue cases in Gombak (2009) ... 12Table 3. Fourier series calculations by using excel for dengue cases in Klang (2009) ..... 16Table 4. Fourier series calculations for dengue cases in Shah Alam (2010) ..................... 20Table 5. Fourier series calculations by using excel for dengue cases in Gombak (2010) . 24Table 6. Fourier series calculations by using excel for dengue cases in Klang (2010) ..... 28Table 7. Fourier series equations on 1st harmonic for 2009 .............................................. 32Table 8. Fourier series equations on 1st harmonic for 2010 .............................................. 32Table 9. Analysis from graph using maple software for 2009 ........................................... 32Table 10. Analysis from graph using maple software for 2010 ......................................... 32 LIST OF FIGURESFigure 1. Graph of dengue cases in Shah Alam, Gombak and Klang for year 2009 ........... 6Figure 2. Graph of dengue cases in Shah Alam, Gombak and Klang for year 2010 ........... 7Figure 3. Fourier series graph plotted for dengue cases in Shah Alam (2009) .................. 11Figure 4. Fourier series graph plotted for dengue cases in Gombak (2009) ...................... 15Figure 5. Fourier series graph plotted for dengue cases in Klang (2009) .......................... 19Figure 6. Fourier series graph plotted for dengue cases in Shah Alam (2010) .................. 23Figure 7. Fourier series graph plotted for dengue cases in Gombak (2010) ...................... 27Figure 8. Fourier series graph plotted for dengue cases in Klang (2010) .......................... 31 iii
  4. 4. ABSTRACTDengue is the most dangerous mosquito virus infection to the human in the world. Up to100 million cases are reported annually and some two billion people are at risk ofinfection in the world. There is no specific cure or medicine to shorten the course ofdengue. 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 chosendistrict and to obtain the highest point for dengue cases in chosen district by referring toFourier series graph plotted. This project focuses on certain districts which hadrecorded 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 nextyear by following the trend line that is generated by Excel. Thus, the alternative that wehave is to transform the graph into a periodic graph using Fourier series, so that thehighest point for the dengue cases can be determined. Fourier series is an expansion of aperiodic function of period which the base is the set of sine functions. Hence, Fourierseries is one of the alternative methods to compare and explain the pattern of denguecases recorded. The result between year 2009 and 2010 show the number of dengue casesseasonally peak at first quarter of year which averagely recorded in period week 7 toweek 14 (February to April). iv
  5. 5. 1. INTRODUCTIONDengue is the most dangerous mosquito virus infection to the human in the world. Angand Li (1999) stated that up to 100 million cases are reported annually and some twobillion people are at risk of infection in the world. Dengue viruses are transmitted fromvector (mosquitoes) to the susceptible human beings by various mosquitoes such asAedes aegypti and Aedes albopictus. From that, the infected person will have a fewsymptoms such as high fever (40°C), chills, headache, pain in the eyes, deep muscle andjoint pains and extreme fatigue. Actually, the infected person will have high fever for twoto four days. Then, the body temperature will drop rapidly and intense sweating takesplaces. But, patient’s body will show up small red bumps. These are a few symptoms thatwill 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 studybecause we must identify and know who is the vector and the host. Basically, dengueviruses are transmitted from the vector (mosquitoes) to the host (humans). Thetransmitted dengue virus process happened by mosquitoes bite during mosquitoes bloodfeeding. The mosquitoes also may carry the virus from one host to another host. Whenthe virus has been transmitted to the host (humans) incubation period will occur. Thedengue viruses multiply during incubation time. After three to five days, the symptomsof dengue will appear and attack patients.There is no specific cure or medicine to shorten the course of dengue. Actually, themedicine 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 andjoint aches, fever and headache. The patient is advice to keep rest in a screened room toprevent 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 candefine the patient is fully recovered and in the best condition (recover person) when thesymptoms had disappeared.The occurrence of dengue in Malaysia had become more serious year to year. TheMinistry of Health Malaysia (2009) stated that dengue has become pandemic. Besidesthat, people did not take this problem as a serious problem. In order to increase thepeople’s sensitivity of dengue, the Ministry of Health has done many activities andcampaign such as advertisement through the television and internet. The activities andcampaign also include involvement of students in primary and secondary schools. Forexample, the competition “AntiAedes Ranjer Ridsect” organized by Sara Lee Company(Ridsect) which cooperated with Ministry of Health Malaysia. 1
  6. 6. In order to analyze the dengue cases which happened in Malaysia for this study, Fourierseries was choose because of its availability to present and show the new perspectiveanalysis of dengue cases. Zill and Cullen (2009) stated that the representation of afunction in the form of a series is widely and frequently used to solve and explain thecommon problem situation.The history of Fourier series started when Bernoulli, D’ Allembert and Euler (1750) hadused and introduced the idea of expanding a function in the form a series to solve theassociated with the vibration of strings. Then, Joseph Fourier who a French physicist,(1768-1830) improved and developed the approach of Fourier series where it wasgenerally useful nowadays. However, the search had done by Joseph Fourier gave impactto 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 theiridea. But, the text of Joseph Forier which Theorie Analytique de la Chaleur (TheAnalytical Theory of Heat) become the source for the modern method in order to solveproblems associated with partial differential equations subject to prescribed boundaryconditions.Nowadays, the application of Fourier series analysis is commonly used in physic andelectrical engineering sector which how frequency associated to a dynamical systems.The text from Joseph Fourier influenced in created electrical component such aselectronic rectifiers. Fourier series also is the best method to analyze the data series suchas dengue cases which useful to compare and determine the dengue cases happened inMalaysia.Angove (2009) analyze the periodic time domain voltage waveform and convert it to thefrequency domain which always uses in electronic communication systems. For examplea waveform usually decomposed into sum of harmonically related sine, cosine waveformand constant which is known as Fourier series.Klingenberg (2005) showed the way to apply and calculate Fourier series analysis byusing Microsoft Excel. Excel generally shows the magnitude versus time is knownwaveform. Klingenberg (2005) done the experiments call for the “harmonic content” of areproduced waveform is a display of the magnitude of the waveform (Y-axis) versus thefrequency (X-axis). In other word, we called it as frequency spectrum and it allowsvisualizing a waveform according to its frequency content.Kvernadzi, Hagstrom and Shapiro (1999) studied about the utilization of the truncatedFourier series and it applies as a tool for the approximation of the points of discontinuitiesand the magnitudes by using integrals. Abas, Daud and Yusuf (2009) studied aboutrainfall by using Fourier series with significant number of harmonics is fitted to themodel’s parameter. The results of their studies showed that statistical properties of theestimated rainfall series were able to match most of those of the historical series. TheFourier series makes the model more parsimony by grabs the seasonal fluctuations withinthe model. 2
  7. 7. The strategy or plan must be systematic. So, modeling how dengue spread amongpopulation localized in a district guides the Ministry of Health Malaysia to prevent theseepidemics become more danger to community. The model were showed the seasonalpattern 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 rightparameters.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 andprevious immunization. Sometimes, factor likes rainfalls, temperature must beconsidered. The virus progression occurs at a daily scale; therefore it must be recorded inweeks or days to be more accurate and precise. So, the prediction and modeling ofdengue 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 andKlang 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 knowthe pattern and trend of dengue cases will be of great significant in reducing the death ratethat will cause by dengue. A good and reliable mathematical modeling about pattern andtrend dengue will help the government to take preventing control and precaution controlto reduce the dengue case in certain time in the future since this disease does not havespecific treatment.The objectives of this study are to know the pattern of dengue cases that happened inchosen district, to obtain the first harmonic equation of Fourier series and compare thepeak value for dengue cases in the chosen district by referring to Fourier series graphplotted. This project focuses on certain districts which had recorded the highest denguecases among district in Malaysia which are Shah Alam, Gombak and Klang. 3
  8. 8. 2. METHODOLOGYSome of Fourier series formula from Zill and Cullen (2009) that are used throughout thisstudy 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 xdxFourier series determined from the coefficient which are a0, an, and bn. Since, we arefocus 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
  9. 9. 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    Lwhere yk is data obtained from the dengue cases and 2L is the period time. Then, wearrange 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 analyzethe trend of dengue cases. 5
  10. 10. 3. IMPLEMENTATIONBefore proceed to Fourier series method, the data of dengue case were plotted by usingExcel 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 2009Figure1 shows that comparison of dengue cases between Shah Alam, Gombak and Klangsince 7 January until 26 December 2009 by graph. From the graph above, it is hard tocompare the pattern between these districts. Thus, it is not accurate if we want togenerate 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
  11. 11. Figure 2. Graph of dengue cases in Shah Alam, Gombak and Klang for year 2010Figure2 shows that comparison of dengue cases between Shah Alam, Gombak and Klangsince 9 January until 8 August 2010 by graph. From the graph above, it is hard tocompare the pattern between these districts. Thus, it is not accurate if we want togenerate 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 inShah Alam, Gombak and Klang is Fourier series. 7
  12. 12. 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
  13. 13. Table 1 shows that the calculations for Fourier series by using Excel. From the table, wecan obtain coefficients of Fourier series which are a0, a1 and b1 where period time, L is25.5 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  216.5415Then, 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
  14. 14. We can write the sum of sine and cosine term, with the same periodic which focus on thefirst 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   0.3579Then,  x  y  80.4412  231.1922 sin   L  0.3579   This equation is plotted by using Maple software to determine the peak value and analyzethe trend of dengue cases. 10
  15. 15. >>> 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 ofweeks. From the graph, it shows that the maximum point or peak point in week 10 with310 dengue cases. However, between week 26 to week 45, the graph shows that theminimum number of case which is zero. It happened because the different or gapbetween actual data for maximum cases and minimum cases is high. Early hypothesisfrom this graph is the highest cases happen in week 10 with 310 cases and the lowestcases happen between week 26 to week 45. 11
  16. 16. 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 0.0000TOTAL 6164 0.0000 0.0000 397.8217 1848.6629 12
  17. 17. Table 2 shows that the calculations for Fourier series by using excel. From the table, wecan obtain coefficients of Fourier series which are a0, a1 and b1 where period time, L is25.5 which is half of 51 (numbers of data). 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  72.4966Then, we arrange the Fourier series as follow: 120.8627  x x  f ( x)   15.6009 cos  72.4966 sin   ... 2  25.5 25.5  13
  18. 18. We can write the sum of sine and cosine term, with the same periodic which focus on thefirst harmonic by calculated the value of c0, c1 and α1. 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   0.2120Then,  x  y  60.4314  74.1562 sin   L  0.2120   This equation is plotted by using Maple software to determine the peak value and analyzethe trend of dengue cases. 14
  19. 19. >>> 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 ofweeks. From the graph, it shows that the maximum point or peak point in week 10 with134 dengue cases. However, from week 32 to week 42, the graph shows that theminimum number of case which is zero. It happened because the different or gapbetween actual data for maximum cases and minimum cases is high. Early hypothesisfrom this graph is the highest cases happen in week 10 with 134 cases and the lowestcases happen between week 32 to week 42. 15
  20. 20. 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
  21. 21. Table 3 shows that the calculations for Fourier series by using excel. From the table, wecan obtain coefficients of Fourier series which are a0, a1 and b1 where period time, L is25.5 which is half of 51 (numbers of data). 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  79.9380Then, we arrange the Fourier series as follow: 51.1765  x x  f ( x)     15.6447 cos  79.9380 sin   ... 2  25.5 25.5  17
  22. 22. We can write the sum of sine and cosine term, with the same periodic which focus on thefirst harmonic by calculated the value of c0, c1 and α1. 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   0.1933Then,  x  y  25.5882  81.4546 sin  L  0.1933    This equation is plotted by using Maple software to determine the peak value and analyzethe trend of dengue cases. 18
  23. 23. >>> 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. Fory-axis represents the number of dengue cases and for x-axis represents the number ofweeks. From the graph, it shows that the maximum point or peak point in week 15 with120 dengue cases. However, from week 30 to week 50, the graph shows that theminimum number of case which is zero. It happened because the different or gapbetween actual data for maximum cases and minimum cases is high. Early hypothesisfrom this graph is the highest cases happen in week 15 with 120 cases and the lowestcases happen between weeks 30 to week 50. 19
  24. 24. 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 0.0000TOTAL 2150 0.0000 0.0000 -24.3271 1118.3775 20
  25. 25. Table 4 shows that the calculations for Fourier series by using excel. From the table, wecan obtain coefficients of Fourier series which are a0, a1 and b1 where period time, L is15.5 which is half of 31 (numbers of data). 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  72.1534Then, we arrange the Fourier series as follow: 69.3548  x x  f ( x)     1.5695 cos  72.1534 sin   ... 2  15.5 15.5  21
  26. 26. We can write the sum of sine and cosine term, with the same periodic which focus on thefirst harmonic by calculated the value of c0, c1 and α1. 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   0.0217Then,  x  y  34.6774  72.1705 sin  L  0.0217    This equation is plotted by using Maple software to determine the peak value and analyzethe trend of dengue cases. 22
  27. 27. >>> 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. They-axis represents the number of dengue cases and the x-axis represents the number ofweeks. From the graph, it shows that the maximum point or peak point in week 9 with120 dengue cases. However, from week 18 to week 28, the graph shows that theminimum number of case which is zero. It happened because the different or gapbetween actual data for maximum cases and minimum cases is high. Early hypothesisfrom this graph is the highest cases happen in week 9 with 120 cases and the lowest caseshappen between week 18 to week 28. 23
  28. 28. 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 0.0000TOTAL 3571 0.0000 0.0000 254.0694 996.4649 24
  29. 29. Table 5 shows that the calculations for Fourier series by using excel. From the table, wecan obtain coefficients of Fourier series which are a0, a1 and b1 where period time, L is15.5 which is half of 31 (numbers of data). 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  64.2881Then, we arrange the Fourier series as follow: 115.1935  x x  f ( x)   16.3981cos  64.2881sin   ... 2  15.5 15.5  25
  30. 30. We can write the sum of sine and cosine term, with the same periodic which focus on thefirst harmonic by calculated the value of c0, c1 and α1. 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   0.2497Then,  x  y  57.5968  66.3448 sin  L  0.2497    This equation is plotted by using Maple software to determine the peak value and analyzethe trend of dengue cases. 26
  31. 31. >>> Figure 7. Fourier series graph plotted for dengue cases in Gombak (2010)Figure 7 shows that Fourier series that plotted with Maple software in first harmonic. They-axis represents the number of dengue cases and the x-axis represents the number ofweeks. From the graph, it shows that the maximum point or peak point in week 7 with120 dengue cases. However, from week 19 to week 24, the graph shows that theminimum number of case which is zero. It happened because the different or gapbetween actual data for maximum cases and minimum cases is high. Early hypothesisfrom this graph is the highest cases happen in week 7 with 120 cases and the lowest caseshappen between week 19 to week 24. 27
  32. 32. 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 0.0000TOTAL 974 0.0000 0.0000 -129.4878 260.0890 28
  33. 33. Table 6 shows that the calculations for Fourier series by using excel. From the table, wecan obtain coefficients of Fourier series which are a0, a1 and b1 where period time, L is15.5 which is half of 31 (numbers of data). 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  16.7799Then, we arrange the Fourier series as follow: 31.4194  x x  f ( x)     8.3541cos  16.7799 sin   ... 2  15.5 15.5  29
  34. 34. We can write the sum of sine and cosine term, with the same periodic which focus on thefirst harmonic by calculated the value of c0, c1 and α1. 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   0.4619Then,  x  y  15.7097  18.7445 sin  L  0.4619    This equation is plotted by using Maple software to determine the peak value and analyzethe trend of dengue cases. 30
  35. 35. >>> Figure 8. Fourier series graph plotted for dengue cases in Klang (2010)Figure 8 shows that Fourier series that plotted with Maple software in first harmonic. They-axis represents the number of dengue cases and the x-axis represents the number ofweeks. From the graph, it shows that the maximum point or peak point in week 10 with35 dengue cases. However, from week 23 to week 28, the graph shows that the minimumnumber of case which is zero. It happened because the different or gap between actualdata for maximum cases and minimum cases is high. Early hypothesis from this graph isthe highest cases happen in week 10 with 35 cases and the lowest cases happen betweenweek 23 to week 28. 31
  36. 36. 4. RESULTS AND DISCUSSIONFrom the findings, it can be noticed that the data of dengue cases in these three districtswhich are Shah Alam, Gombak and Klang is distributed fluctuation. It is difficult todetermine and predict the dengue cases for the next year by following the trend line that isgenerated 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
  37. 37. The equation of Fourier series on first harmonic for dengue cases are shown in Table 7and Table 8. For the year 2009, dengue cases seasonally peak between period week 10 to14 (14 March 2009 until 11 April 2009) averagely recorded between 100 and 300 casesper week. It also shows that dengue cases dengue cases slowly decrease for chosendistrict at the end of year 2009. For the year 2010, dengue cases seasonally peak betweenperiod week 7 to week 10 (20 February 2010 until 13 March 2010) averagely recordedbetween 35 and 120 cases per week. It also shows that dengue cases dengue cases slowlydecrease 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 casesseasonally peak at first quarter of year which averagely recorded in period week 7 toweek 14 (February to April). Then, the dengue cases will reduce slowly in the thirdquarter of the year.5. CONCLUSIONS AND RECOMMENDATIONSDengue is one of the diseases with no specific treatment or immunizations. Thus, thepreventive 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 yearwhich averagely recorded in period week 7 to week 14 (February to April). Shah Alamrecorded the highest dengue cases in 2009 which 310 cases in week 10 compared toKlang which recorded 110. In year 2010, Shah Alam and Klang show drastic decrease thenumber of cases which 120 and 35 cases respectively. However, Gombak did not recordthe 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 focusmore on first quarter of the year (February until April) every year to reduce dengue casesbecause 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 seasonalpeak can be determined further. This model can be explored further by comparing thedengue cases recorded with climatic variability that is rainfalls, temperature and vaporpressure in those selected districts in Selangor. Comparison can also be done betweenstates in Malaysia. 33
  38. 38. REFERENCESAbas 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 ofMathematics, 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: Stakesand Pitfalls.Klingenberg, L. (2005). Frequency Domain Using Excel. San Francisco State UniversitySchool of Engineering.Kvernadze, G., Hagstrom, T., and Shapiro, H. (2000). Detecting the Singularities of aFunction of VP Class by its Integrated Fourier Series. Computers and Mathematics withApplications, 39, 25-43.McNeal, J. D. and Zeytuncu, Y. U. (2006). A note on rearrangement of Fourierseries. J. Math. Anal. Appl. 323 (2006) 1348–1353.Nuraini, N., Soewono, E. & Sidarto, K.A. (2006). Mathematical Model of DengueDisease Transmission with Severe DHF Compartment, Bulletin of the MalaysianMathematical Sciences Society, 30, 143-157.Pongsumpun, P. & Tang, I.M. (2001). A Realistic Age Structure Transmission Model forDengue 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-ValueProblems, 7th Edition, International Student Edition. 34

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