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Parameter Estimation of Sir Model

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TECHNICAL REPORT
PARAMETER ESTIMATION OF SIR MODEL FOR THE SPREAD OF DENGUE DISEASE

HAZWANI BINTI AHMAD (2009224096 - TR12/20) NURUL HAMIZAH BINTI MOKHTAR (2009283442 - TR12/20) SITI ATIQAH BINTI RAMLI (2009652882 - TR12/20)

UNIVERSITI TEKNOLOGI MARA

TECHNICAL REPORT
PARAMETER ESTIMATION OF SIR MODEL FOR THE SPREAD OF DENGUE DISEASE

HAZWANI BINTI AHMAD (2009224096 - TR12/20) NURUL HAMIZAH BINTI MOKHTAR (2009283442 - TR12/20) SITI ATIQAH BINTI RAMLI (2009652882 - TR12/20)

Report submitted in partial fulfilment of the requirement for the degree of Bachelor of Science (Hons.) ( Computational Mathematics) Center of Mathematics Studies Faculty of Computer and Mathematical Sciences

JANUARY 2012

ACKNOWLEDGEMENTS

First of all, we are grateful to ALLAH S.W.T for his mercy and guidance in giving us full-strength to complete this “Projek Ilmiah Tahun Akhir (PITA)” that really tested our abilities mentally and physically. Even facing with some difficulties in

completing this project, we still manage to complete it. We also want to show our thanks to those who have contributed their leadership, guidance and support for the completion of this project. Special appreciation to the our supervisor, Puan Hanimah Binti Basri for all of her encouragement, guidance and support in helping us to understand and enable to complete this project from initial of this project until the final level. Her ideas and suggestions to this project are very valuable at each stage of our work. This report would never have finished without her constructive comments, great advice and continuous support. We would like to express our gratitude to Dr. Fuziyah Binti Ishak who gave us full assistance and information to finish this report. We would like to thank her also for her guidance and great support either in class or outside class. In addition, grateful acknowledgement to the lecturers and our friends of the Faculty of Computer and Mathematical Science who are play the important roles in this study. Thank you very much for their help, assistance, support and advices to complete our project. Finally, we would like to convey our warmest gratitude to our family especially our loving parents for their love, emotional support and encouragement during completing this project. Thank you.

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TABLE OF CONTENTS
ACKNOWLEDGEMENTS ........................................................................................... i TABLE OF CONTENTS............................................................................................. iii LIST OF TABLES ....................................................................................................... iv LIST OF FIGURES ..................................................................................................... iv ABSTRACT .................................................................................................................. v 1. INTRODUCTION .................................................................................................. 1 2. METHODOLOGY ................................................................................................. 6 2.1 Data and Data Collection ................................................................................ 6 2.2 Method ............................................................................................................11 2.2.1 Modeling SIR model ........................................................................... 11 2.2.2 Solving the SIR equations ................................................................... 12

3. IMPLEMENTATION ...........................................................................................19 4. RESULTS AND DISCUSSIONS ...........................................................................25 4.1 Analysis of Data..............................................................................................25 4.2 Analysis of Graph...........................................................................................27 5. CONCLUSION ......................................................................................................35 6. RECOMMENDATIONS AND FUTURE STUDY ...............................................35 7. REFERENCES ......................................................................................................36

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LIST OF TABLES
Table 1. Number of dengue cases at MBPJ in 2007 ...................................................... 7 Table 2. Number of dengue cases at MPK in 2007 ........................................................ 8 Table 3. Number of dengue cases at MBPJ in 2008 ....................................................... 9 Table 4. Number of dengue cases at MPK in 2008 ...................................................... 10 Table 5. Value of alpha and beta at MBPJ in 2007 ...................................................... 13 Table 6. Value of alpha and beta at MPK in 2007........................................................ 14 Table 7. Value of alpha and beta at MBPJ in 2008 ...................................................... 15 Table 8. Value of alpha and beta at MBK in 2008 ....................................................... 16 Table 9. Summary of the Sum and Average for alpha and beta ................................... 17 Table 10. Summary of equilibrium point and stability point, λ1 and λ2 ......................... 25 Table 11. Number of infected person of actual and estimation data for MBPJ in 2007. 31 Table 12. Number of infected person of actual and estimation data for MPK in 2007 .. 32 Table 13. Number of infected person of actual and estimation data for MBPJ in 2008. 33 Table 14. Number of infected person of actual and estimation data for MPK in 2008 .. 34

LIST OF FIGURES
Figure 1. Aedes aegypti mosquito ................................................................................. 1 Figure 2. Aedes albopictus mosquito ............................................................................. 1 Figure 3. Graph of S’,I’,R’ for MBPJ 2007 ................................................................. 28 Figure 4. Graph of S’,I’,R’ for MPK 2007................................................................... 28 Figure 5. Graph of S’,I’,R’ for MBPJ 2008 ................................................................. 29 Figure 6. Graph of S’,I’,R’ for MPK 2008................................................................... 29

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ABSTRACT
In this study we will look at the SIR model for the mathematical modeling of dengue disease because it is important to understand and predict this spread disease. It is because it can reduce the number of persons who has infected. This SIR model is applied using the Euler method to estimate the parameter α and β. Our main scope that takes into account is Selangor which we are specific the scope only in Petaling Jaya City Council (MBPJ) and Klang Municipal Council (MPK). The objectives of this project are using the SIR model of dengue disease in MBPJ and MPK in Selangor by determining the parameter estimation of α, β of the SIR model for dengue disease and also from that determine the coefficient of the Jacobian matrix, λ and reproduction ratio, R0. The values of λ and R0 are used to determine stability of equilibrium points.

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1.

INTRODUCTION

Dengue is a human disease caused by viruses that are transmitted by mosquitoes. Dengue is an infectious disease that is dangerous and can cause of death. Dengue can occur to anybody either adults or children resulting from the bite of Aedes mosquito. There are two types of Aedes mosquito that cause dengue which are Aedes aegypti and Aedes albopictus. This disease is classified into two forms; Dengue Fever (DF), Dengue Hemorrhagic Fever (DHF) which may evolve toward a severe form known as Dengue Shock Syndrome (DSS). According Derouich et al. (2003), the problem with dengue is caused by four distinct serotypes known as DEN1, DEN2, DEN3 and DEN4.

Figure 1. The Aedes aegypti mosquito mosquito

Figure 2. Aedes albopictus

There are several symptoms of dengue such as headache, fever, exhaustion, joint and muscle pain, swollen glands (lymphadenopathy), and rash. There are no specific antiviral medicines for dengue, but when someone has been detected has dengue fever it suggests to maintain hydration and avoid using the acetylsalicylic acid (e.g. Aspirin) and non steroidal anti inflammatory drugs (e.g. Ibuprofen). The bites of the female striped Aedes aegypti mosquito (vector) cause transmit the dengue viruses to human (host). These mosquitoes easily flourish during the rainy seasons but it also can flourish in fresh water such as water that is stored in plastic bags, cans, flowerpots and old tires. During probing and blood feeding, the dengue virus is transmitted to its host. These dengue viruses may carried by mosquito from one host to the other host. As mention earlier, a mosquito bite can cause the disease.

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When the viruses have been transmitted to the human host, it can be said that there is occur the incubation period. During this period, the dengue viruses multiply. The characteristics of dengue appear after period ranges from 3 to 15 days (usually lasting for 5-8 days). The risk of being bitten is highest during the early in the morning and in the late afternoon before sunset. It is because the mosquito is most active during that period. According to research that had been done by Derouich et al. (2003) dengue fever hitting countries in Southeast Asia with warm tropical and warm climates. Dengue fever occurs in two forms; the Dengue Fever (DF) or classic dengue and the Dengue Hemorrhagic Fever (DHF) which may know as Dengue Shock Syndrome (DSS) that transmitted by the mosquito of the genus Aedes to the people. Derouich also discuss about the prevention of dengue epidemic that use vector control through environmental situation. This prevention is to eliminate resting places of larval to delay the spread of the epidemic. Recently, there has been a notable increase in dengue fever and dengue hemorrhagic fever cases in both the very young and in aged adults and also, dengue from pregnant women had been increasingly reported (Pongsumpun and Kongnuy, 2007). There is 30 percent of dengue infection is reported in patients who are more than 15 years old. Some pregnant women are risking being susceptible to dengue. If they experience dengue infection, then they can transmit the dengue viruses to their babies. Therefore, the method of an ultra low volume (ULV) amount of insecticides could be used to co people so that the basic reproduction rate will reduce to below one. Thus, we can reduce the outbreak of the disease. Nowadays, there are many mathematical models are used in the study the infection disease epidemiology between dengue transmission in human and vector population. Its use for giving better detection, prevention and control program by take care the comparing, planning, implementing and evaluating the transmission of the dengue vector. In our research about dengue disease, we are applying the SIR model to our research. In the SIR model, assume that at any given time t, a fixed population can be split into three classes; class of susceptible persons (those naive to the disease), class of infectious persons (those with the disease), and class of recovered persons (those who 2

had the disease and are as a result immune). In all three classes, the total number of people is assumed to be constant through time and the basic SIR models assume fixed population size N, with no births or deaths from causes other than the disease itself. The model referred as classical SIR is given by the system of differential equations where S is who susceptible to disease but not yet infected, I is whose contact with susceptible results in an infection and R is referring to individual recovered from the population with permanent immunity. One of mathematical model that's being used to determine the parameter estimation of the SIR model is The Multistage Adomian Decomposition Method (MADM). The MADM is a decomposition method used in approximating the solution of a system of ordinary differential equations. Research made by Nuraini et al. (2010) state that the possibility of this method being used to estimate the parameter in an SIR model. Pongsumpun and Tang (2001) were studied about Susceptible, Infected, and Recovered (SIR) model to explain the transmission of dengue hemorrhagic fever (DHF) is influenced by age structure. DHF was first occurring in the Philippines and the large last of outbreak occur in 1998 in Thailand. In the mathematical model studied that subscript ‘i’ refers to age structure was used in the SIR i.e. Si, Ii and Ri. But the studied shown that the highest influence of dengue is respective region rather than age structure. In the World Health Organization (1999) monograph, it is

mentioned that the control and prevention of dengue fever and DHF should center on the eradication of the transmitting vector in the absence of an effective vaccine against the dengue virus. The basic reproduction ratio of the epidemic, R0 is defined as the expected number of secondary infections generated by a single, typical infection in the completely susceptible population. Five fundamental insights into the dynamics of an infectious disease from the R0 are represented as the R0 is the threshold parameter that will determine whether or not there will be an epidemic, R0 determines the initial rate of the increase of an epidemic, R0 determines the final size of the epidemic, R0 determines the endemic equilibrium fraction of susceptible in the population and R0 determines the critical vaccination threshold. Based on Pongsumpun and Tang (2001), the reproduction number, R0 is related to the prevention program. By reducing the value of R0, it will lower the biting rate or use 3

the mosquito net for protection. The prevention program should have to be a continuing one. Research made by Boëlle et al (2009), it stated that there are two parameters must be estimated which are the reproduction ratio (R) and generation interval. The average number of secondary cases per primary case will be used to measure the reproduction ratio meanwhile the average time between infection in a primary case and its secondary cases is used to measure the generation interval. The reproduction ratio will affect the efficacy of public health interventions. It means that, the larger the reproduction ratio is, public health interventions also will increase. This parameter can be used to determine the recovered case and estimate any dengue virus in the future so that prevention way can be takes early. Besides using the mathematical model to determine and prevent the mosquitotransmitted disease, there is some other research that's done not based on the mathematical model. Bianca (2009) state that, Scott O’Neill, an entomologist at the University of Queensland have been working on one such method-creating a vaccine for the mosquito rather than for the people and can be developed over very large areas, inexpensive also without treating humans at all. The research focus on creating a naturally occurring bacterium which destroys the ability of infected mosquitoes to function as carries. Wolbachia-certain example of strain of bacteria- infects up to 70% of insect species, and if infected, the mosquito may not live long enough for its dengue passenger to incubate and move on. Our scope in this study is in Selangor which we more specific on Petaling Jaya City Council (MBPJ) and Klang Municipal Council (MPK). Area of Petaling Jaya City Council is approximately 97.2 km2, making it the fastest growing city in the state. Petaling Jaya is located in the Petaling district, which is the largest and most developed district in the state. Petaling Jaya is the second city in the state after the Shah Alam and a city that is well planned. The areas under Klang Municipal Council are Port Klang, Pandamaran, South Klang, North Klang, Meru and Kapar. To best our knowledge, we do research about spread of dengue in MBPJ and MPK since no research before done about spread of dengue in MBPJ and MPK. The impact of dengue disease occurs to population of human such as changes in death rates in MBPJ and MPK of Selangor because the dengue disease can causes the death. Varieties of

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mathematical tool are useful to link mathematical model and data to give better understanding and prediction of the spread diseases. . The objectives of this project are to estimate the value of α and β of the SIR model for dengue disease and to determine the stability of the equilibrium points through value of coefficient of Jacobian matrix, λ and reproduction ratio, R0.

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2.
2.1

METHODOLOGY
Data and Data Collection

Data used for this research is secondary data, which is obtained from the Ministry of Health, Malaysia. The data is about the number of cases of dengue which is meant the population of people that infected by dengue disease in Selangor. From that data the state that takes into account is Selangor which is in year 2007 and 2008 while the council that chosen are Petaling Jaya City Council (MBPJ) and Klang Municipal Council (MPK). Besides that, we also used primary data to collect the data of population size. The primary data that we used is by calling the Klang Municipal Council and Petaling Jaya City Council and we gathered the data of population in the year 2007 and 2008 respectively. The population size that obtained are as follows: for the Petaling Jaya City Council are 417030 in 2007 and 505300 in 2008, meanwhile the population for the Klang Municipal Council are 693956 in 2007 and 725192 in 2008. Then, based on data obtained above we know that the data already have is cases of the infected people, I and the initial population of susceptible person, S0 and from that information we can get the number of recovered people, R and susceptible person, S. To get both numbers of cases, we derive the equation as equation (1) and equation (2). The equation (1) shows how to get the number of recovered people meanwhile the equation (2) shows how to get the susceptible person, Rn +1 = I n +1 + Rn , S n +1 = S n − Rn +1 − I n +1 . (1) (2)

Therefore, we use equation (1) and equation (2) in tables below for dengue disease in Petaling Jaya City Council (MBPJ) and Klang Municipal Council (MPK) respectively for 2007 and 2008. Table 1, Table 2, Table 3, and Table 4 below are about the data of number of cases in time (t) in weeks which are susceptible (S), infection (I) and recovered (R) person for dengue disease where Table 1 at Petaling Jaya City Council (MBPJ) in 2007, Table 2

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at Klang Municipal Council (MPK) in 2007, Table 3 at Petaling Jaya City Council (MBPJ) in 2008 and Table 4 at Klang Municipal Council (MPK) in 2008. Table 1. Number of Dengue Cases at MBPJ in 2007 t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 S 416961 416898 416813 416738 416681 416623 416578 416578 416563 416558 416522 416487 416457 416412 416382 416342 416306 416275 416229 416169 416124 416070 416028 415981 415937 415887 415853 I 69 63 85 75 57 58 45 0 15 5 36 35 30 45 30 40 36 31 46 60 45 54 42 47 44 50 34 R 0 69 132 217 292 349 407 452 452 467 472 508 543 573 618 648 688 724 755 801 861 906 960 1002 1049 1093 1143 t 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 S 415816 415777 415749 415724 415688 415644 415594 415560 415518 415470 415401 415278 415228 415143 415104 415048 415048 414990 414934 414887 414849 414820 414820 414820 414820 414820 414820 I 37 39 28 25 36 44 50 34 42 48 69 123 50 85 39 56 0 58 56 47 38 29 0 0 0 0 0 R 1177 1214 1253 1281 1306 1342 1386 1436 1470 1512 1560 1629 1752 1802 1887 1926 1982 1982 2040 2096 2143 2181 2210 2210 2210 2210 2210

As we can perceive from Table 1 above, the number of infected persons is only until week 49 while the rest of the week is no person have been infected.

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Table 2. Number of Dengue Cases at MPK in 2007 t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 S 693947 693941 693933 693926 693914 693910 693906 693900 693896 693896 693890 693883 693876 693873 693873 693869 693865 693859 693852 693835 693825 693821 693817 693811 693803 693796 693785 I 9 6 8 7 12 4 4 6 4 0 6 7 7 3 0 4 4 6 7 17 10 4 4 6 8 7 11 R 0 9 15 23 30 42 46 50 56 60 60 66 73 80 83 83 87 91 97 104 121 131 135 139 145 153 160 t 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 S 693774 693774 693768 693763 693757 693747 693746 693736 693734 693731 693731 693728 693724 693719 693717 693714 693714 693710 693706 693706 693696 693690 693690 693690 693690 693690 693690 I 11 0 6 5 6 10 1 10 2 3 0 3 4 5 2 3 0 4 4 0 10 6 0 0 0 0 0 R 171 182 182 188 193 199 209 210 220 222 225 225 228 232 237 239 242 242 246 250 250 260 266 266 266 266 266

Based on Table 2 above there are several weeks that have person not infected which are on week 10, 15, 29, 38, 44, 47 and week 50 until 54. The number of infected persons for an entire week at MPK in 2007 is below 20 persons. This number of infected is less than MBPJ 2007 which is number of infected persons for an entire week at MBPJ in 2007 is below 130 persons.

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Table 3. Number of Dengue Cases at MBPJ in 2008 t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 S 505282 505267 505247 505228 505213 505191 505168 505135 505101 505069 505046 505023 504992 504955 504930 504907 504885 504852 504814 504769 504734 504700 504676 504647 504612 504580 504534 I 18 15 20 19 15 22 23 33 34 32 23 23 31 37 25 23 22 33 38 45 35 34 24 29 35 32 46 R 0 18 33 53 72 87 109 132 165 199 231 254 277 308 345 370 393 415 448 486 531 566 600 624 653 688 720 t 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 S 504496 504470 504433 504407 504364 504336 504310 504288 504266 504246 504222 504198 504186 504161 504137 504123 504107 504092 504072 504049 504021 503987 503963 503935 503908 503871 503871 I 38 26 37 26 43 28 26 22 22 20 24 24 12 25 24 14 16 15 20 23 28 34 24 28 27 37 0 R 766 804 830 867 893 936 964 990 1012 1034 1054 1078 1102 1114 1139 1163 1177 1193 1208 1228 1251 1279 1313 1337 1365 1392 1429

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Table 4. Number of Dengue Cases at MPK in 2008 t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 S 725112 725041 724980 724911 724853 724800 724735 724685 724643 724590 724554 724509 724463 724402 724339 724279 724200 724125 724056 723977 723915 723850 723787 723716 723654 723580 723506 I 80 71 61 69 58 53 65 50 42 53 36 45 46 61 63 60 79 75 69 79 62 65 63 71 62 74 74 R 0 80 151 212 281 339 392 457 507 549 602 638 683 729 790 853 913 992 1067 1136 1215 1277 1342 1405 1476 1538 1612 t 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 S 723438 723387 723335 723283 723231 723167 723110 723059 723017 722972 722909 722859 722820 722772 722723 722691 722651 722616 722560 722513 722466 722395 722341 722251 722174 722124 722124 I 68 51 52 52 52 64 57 51 42 45 63 50 39 48 49 32 40 35 56 47 47 71 54 90 77 50 0 R 1686 1754 1805 1857 1909 1961 2025 2082 2133 2175 2220 2283 2333 2372 2420 2469 2501 2541 2576 2632 2679 2726 2797 2851 2941 3018 3068

From Table 3 and Table 4 above, we can see that on week 54 only no person has been infected. Number of infected persons for entire week 2008 at MPK is below 100 persons which are more than half of MBPJ.

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2.2

Method

2.2.1 Modeling SIR model

Initially, this model is appropriate one for us to use under several assumptions: i) ii) The population of MBPJ and MPK is fixed. The only way a person can leave the susceptible group is to become infected persons and the way a person can leave the infected group is to recover from the dengue disease. A person who has received immunity is recovering. iii) The probability of being infected is not influenced by age, sex, social status and race. iv) There is no inherited immunity.

Johnson (2009) stated the SIR model in basic notation: • • • • S(t) is number of susceptible persons at time t. I(t) is the number of infected persons at time t. R(t) is number of recovered persons at time t. N is the total population size.

Thus, the assumptions and basic notation lead us to a set of system differential equations as follows
S ' = − β ⋅ I (t ) ⋅ S (t ), I ' = β ⋅ I (t ) ⋅ S (t ) − α ⋅ I (t ), R ' = α ⋅ I (t ),

(3) (4) (5)

where beta, β is the infection rate person in a time period (with β greater or equal to zero), and alpha, α is the recovered rate (with α greater or equal to zero) with initial conditions S (0) = S 0 > 0, I (0) = I 0 > 0, R0 ≥ 0 and constant population is

S (t ) + I (t ) + R(t ) = N .

(6)

Based on equation (5) and equation (6), we can discover how the different groups will produce as t→∞ we can see from equation (3), that the susceptible group will decrease over time and approach to zero. Then, from equation (5) we can see that the

recovered group increase and will approach N over time.

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2.2.2 Solving the SIR equation

By applying Euler’s method of system, we can solve for alpha, α and beta, β of SIR model. In ways to obtain small error difference, we may assume a very small step size for the time intervals such as 10 −6 . Then, the Euler method of differential equations is

y1 − y 0 = y ' ⋅ ( x1 − x 0 ), y1 = y 0 + y ' ⋅ ( x1 − x0 ) Next, substitute equation (7) into equations (3), (4) and (5) and it becomes: S n +1 = S n − β ⋅ I n ⋅ S n ∆t , I n +1 = I n + ( β ⋅ S n − α ) ⋅ I n ∆t , Rn+1 = Rn + α ⋅ I n ∆t. (8) (9) (10) (7)

After that, we solve the equation (8) to get a value of β. Thus, the mathematical model of α and β are as follow

β=

S n − S n+1 Sn ⋅ I n

and

α=

1 duration of infection

where the duration of infection that take into account is 7 days. Based on Table 5, 6, 7 and 8 below the value of α and β is taking until four decimal places using data of susceptible and infected people in Table 1, 2, 3 and 4. So the value of α is

α=

1 = 0.1429 and substitutes the value of α and β into the equation (3), (4) and (5). 7

Table 5, table 6, table 7, and table 8 below are about the values of alpha (α) and beta (β) where α is recovered rate while β is infection rate dengue at the time (t) in weeks.

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Table 5 is about the value of α and β at Petaling Jaya City Council (MBPJ) in 2007 using the data of susceptible (S) and infection (I) persons in Table 1.

Table 5. Value of alpha and beta at MBPJ in 2007 t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 β 1.8072 × 10-6 2.6708 × 10-6 1.7469 × 10-6 1.5049 × 10-6 2.0151 × 10-6 1.5367 × 10-6 0 6.6028 × 10-7 1.4262 × 10-5 1.9260 × 10-6 1.6981 × 10-6 2.9719 × 10-6 1.3210 × 10-6 2.6421 × 10-6 1.7835 × 10-6 1.7066 × 10-6 2.9410 × 10-6 2.5854 × 10-6 1.4868 × 10-6 2.3791 × 10-6 1.5422 × 10-6 2.2190 × 10-6 1.8566 × 10-6 2.2538 × 10-6 1.3488 × 10-6 2.1587 × 10-6 α 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 t 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 β 2.0910 × 10-6 1.4244 × 10-6 1.7715 × 10-6 2.8572 × 10-6 2.4252 × 10-6 2.2551 × 10-6 1.3496 × 10-6 2.4518 × 10-6 2.2685 × 10-6 2.8537 × 10-6 3.5392 × 10-6 8.0728 × 10-7 3.3764 × 10-6 9.1143 × 10-7 2.8525 × 10-6 0 1.9185 × 10-6 1.6679 × 10-6 1.6069 × 10-6 1.5169 × 10-6 0 α 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7

After substituting the average value of α and β, it will produce the equation below;
S ' = −2.6041 × 10 −6 ⋅ I ⋅ S , I ' = 2.6041 × 10 −6 ⋅ I ⋅ S − 0.1429 ⋅ I , R = 0.1429 ⋅ I .
'

(11)

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Table 6 is about the value of α and β at Klang Municipal Council (MPK) in 2007 using susceptible (S) and infection (I) persons in Table 2.

Table 6. Value of alpha and beta at MPK in 2007 t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 β 9.60688×10 1.92139×10-6 1.26093×10-6 2.47042×10-6 4.80367×10-7 1.44111×10-6 2.16168×10-6 9.60753×10-7 0 1.68134×10-6 1.44117×10-6 6.17648×10-7 0 1.44119×10-6 2.1618×10-6 1.68142×10-6 3.50013×10-6 8.47803×10-7 5.76514×10-7 1.44129×10-6 2.16195×10-6 1.92175×10-6 1.26116×10-6 2.26497×10-6 1.44137×10-6
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α 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7

t 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

β 0 1.20117×10-6 1.7297×10-6 2.40238×10-6 1.44145×10-7 1.44145×10-6 2.88294×10-7 2.16221×10-6 0 1.92198×10-6 1.80187×10-6 5.76602×10-7 2.16227×10-6 0 1.44152×10-6 0 8.64932×10-7 0 -

α 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 -

After substituting the average value of α and β, it will produce the equation below;
S ' = −1.56 × 10 −6 ⋅ I ⋅ S , I ' = 1.56 × 10 −6 ⋅ I ⋅ S − 0.1429 ⋅ I , R = 0.1429 ⋅ I .
'

(12)

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Table 7 below is about the value of α and β at Petaling Jaya City Council (MBPJ) on 2008 using susceptible (S) and infection (I) persons in Table 3.

Table 7. Value of alpha and beta at MBPJ in 2008 t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 β 1.9983 ×10-6 3.1975 ×10-6 2.2783 ×10-6 1.8934 ×10-6 3.5177 ×10-6 2.5076 ×10-6 3.4416 ×10-6 2.4716 ×10-6 2.2579 ×10-6 1.7245 ×10-6 2.3994 ×10-6 3.2341 ×10-6 2.8641 ×10-6 1.6216 ×10-6 2.2080 ×10-6 2.2958 ×10-6 3.6005 ×10-6 2.7642 ×10-6 2.8429 ×10-6 1.8674 ×10-6 2.3326 ×10-6 1.6950 ×10-6 2.9018 ×10-6 2.8986 ×10-6 2.1960 ×10-6 3.4530 ×10-6 1.9845 ×10-6 α 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 t 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 β 1.6438 ×10-6 3.4192 ×10-6 1.6885 ×10-6 3.9743 ×10-6 1.5649 ×10-6 2.2318 ×10-6 2.0338 ×10-6 2.4037 ×10-6 2.1853 ×10-6 2.8848 ×10-6 2.4041 ×10-6 1.2021 ×10-6 5.0090 ×10-6 2.3083 ×10-6 1.4027 ×10-6 2.7482 ×10-6 2.2545 ×10-6 3.2065 ×10-6 2.7657 ×10-6 2.9280 ×10-6 2.9207 ×10-6 1.6980 ×10-6 2.8066 ×10-6 2.3199 ×10-6 3.2970 ×10-6 0 α 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 -

After substituting the average value of α and β, it will produce the equation below;
S ' = −2.0510 × 10 −6 ⋅ I ⋅ S , I ' = 2.0510 × 10 − 6 ⋅ I ⋅ S − 0.1429 ⋅ I , R ' = 0.1429 ⋅ I .

(13)

15

Table 8 below is about the value of α and β at Klang Municipal Council (MPK) on 2008 using susceptible (S) and infection (I) persons in Table 4.

Table 8. Value of alpha and beta at MPK in 2008 t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 β 1.22395×10-6 1.18497×10-6 1.56025×10-6 1.15956×10-6 1.26066×10-6 1.69207×10-6 1.0614×10-6 1.15912×10-6 1.74142×10-6 9.3742×10-7 1.7252×10-6 1.41092×10-6 1.83044×10-6 1.42571×10-6 1.31483×10-6 1.8179×10-6 1.31092×10-6 1.2705×10-6 1.58127×10-6 1.08403×10-6 1.44822×10-6 1.33899×10-6 1.55707×10-6 1.20661×10-6 1.64934×10-6 1.38202×10-6 1.27009×10-6 α 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 t 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 β 1.03672×10-6 1.40949×10-6 1.38249×10-6 1.38258×10-6 1.70177×10-6 1.23156×10-6 1.23735×10-6 1.13895×10-6 1.48189×10-6 1.93645×10-6 1.09786×10-6 1.07905×10-6 1.70273×10-6 1.41239×10-6 9.03612×10-7 1.72965×10-6 1.21082×10-6 2.21418×10-6 1.16154×10-6 1.38406×10-6 2.09095×10-6 1.05284×10-6 2.30731×10-6 1.18457×10-6 8.99161×10-7 0 α 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 1/7 -

After substituting the average value of α and β, it will produce the equation below;
S ' = −1.3769 × 10 −6 ⋅ I ⋅ S , I ' = 1.3769 × 10 −6 ⋅ I ⋅ S − 0.1429 ⋅ I , R = 0.1429 ⋅ I .
'

(14)

16

From that, we summarize the value of summation and average for α and β in Table 9.
Table 9. Summary of the Total and Average for alpha and beta.
Year 2007 Council MBPJ MPK 2008 MBPJ MPK Total of β 1.2239×10
-4

Average of β 2.6041×10
-6

Total of α 0.1429 0.1429 0.1429 0.1429

Average of α 0.1429 0.1429 0.1429 0.1429

6.7200×10-5 1.0870×10
-4 -5

1.5600×10-6 2.0510×10 1.3769×10
-6 -6

7.2975×10

In general, to determine the stability value for both municipal and city council we should find the value of coefficient of Jacobian matrix, λ. From the value of λ , we can determine the stability of the equilibrium point. Before we proceed with the general analytical solution of value λ, we show how to get the value of equilibrium point in general. First, we take two equations which are equation (3) and (4) and let both of them equal to zero as follow
0 = −β ⋅ I ⋅ S , 0 = β ⋅ I ⋅ S − α ⋅ I.

(15) (16)

From equation (15) and (16), S is equal to zero and the value I is also equal to zero, hence the first point is (0, 0). To determine the second point we consider the equation (16) as below;

β ⋅ S ⋅ I − α ⋅ I = 0, β ⋅ S ⋅ I = α ⋅ I, α ⋅I S= , β ⋅I α S= . β

(17)

17

Then substitute equation (17) into the equation (15);

− β ⋅ S ⋅ I = 0, − β ⋅ ( ) ⋅ I = 0, I= 0 ,

α β

α − β ⋅( ) β

I = 0,
α  hence we get the second point which is  ,0  . β   

After this, we show the general analytical solution to get the value of λ of the Jacobian matrix. Firstly, we consider the equation (3) in form dS = P ( s, i ) dt , dI = Q ( s, i ) dt dS dI = 0 and = 0, dt dt

(18)

and let

Secondly, from the equation (18) we let J be the Jacobian matrix where
 ∂P  J =  ∂s  ∂Q   ∂s ∂P   ∂i  and let A be the Jacobian matrix evaluated at s = s (0) and i = i (0) ∂Q   ∂i  ∂P   ∂i  | ( s ( 0 ), i ( 0 )) ∂Q   ∂i  .

 ∂P  A = J | ( s ( 0 ), i ( 0 )) =  ∂s  ∂Q   ∂s

Lastly, to determine the value of λ, let determinant of A − λI equal to zero. Hence, we used this general analytical solution to find the value of λ for both councils in 2007 and 2008.

18

3.

IMPLEMENTATION

The analytical solution to get the value of λ for both councils are show here with the equilibrium point that already compute using software of Maple. First is an analytical solution for MBPJ 2007 based on the equation (11): dS = −2.6041 × 10 −6 ⋅ I ⋅ S , dt dI = 2.6041 × 10 −6 ⋅ I ⋅ S − 0.1429 ⋅ I , dt

this implies

0 = −2.6041 × 10 −6 ⋅ I ⋅ S , 0 = 2.6041 × 10 −6 ⋅ I ⋅ S − 0.1429 ⋅ I , − 0.0000026041i  − 0.0000026041i  then J =   0.0000026041i 0.0000026041s − 0.1429    

We solve at the point (0, 0) as follow 0 0  A = J | (0,0) =   0 − 0.1429     then it becomes 0 0  1 0 − λ A − λI =   0 − 0.1429  − λ  0 1  = 0        −λ 0 0 = 0, − 0.1429 − λ 0 − 0.1429 − λ ,

− λ (−1 − λ ) = 0, and the values of λ are

λ = 0 and

− 0.1429 − λ = 0

λ = −0.1429 .

19

Next, we solve at the point (54875, 0) as follows

 0 − 0.1428999875  A = J | {54875, 0 ) =   0 0.0000000125     then it becomes  0 − 0.1428999875   1 0  − λ − 0.1428999875  A − λI =   0 0.0000000125  − λ  0 1  = 0 0.0000000125 − λ       − λ − 0.1428999875 = 0, 0 0.0000000125 − λ − λ (0.0000000125 − λ ) = 0, and the values of λ are

λ =0

and

0.0000000125 − λ = 0

λ = 0.0000000125.

Second is an analytical solution for MPK 2007 based on the equation (12): dS = −1.56 × 10 −6 ⋅ I ⋅ S , dt dI = 1.56 × 10 −6 ⋅ I ⋅ S − 0.142857 ⋅ I , dt

this implies 0 = −1.56 × 10 −6 ⋅ I ⋅ S , 0 = 1.56 × 10 −6 ⋅ I ⋅ S − 0.142857 ⋅ I .

 − 0.00000156 ⋅ I Then the Jacobian matrix, J =   0.00000156 ⋅ I  Solving the point at (0,0) as below 0 0  A = J | (0,0) =   0 − 0.142857    

− 0.00000156 ⋅ S   0.00000156 ⋅ S − 0.142857  

20

then it becomes 0 0  1 0 − λ  A − λI =   0 − 0.142857  − λ  0 1  = 0       −λ 0 = 0, 0 − 0.142857 − λ − λ (−0.142857 − λ ) = 0. Thus the values of λ are 0 − 0.142857 − λ

λ =0

and

− 0.142857 − λ = 0

λ = −0.142857.

Solving the point at (91575,0) as below  0 − 0.142857  A = J | {91575, 0 ) =   0 − 0.000043     then it becomes  0 − 0.142857  1 0 − λ A − λI =   0 − 0.000043 − λ  0 1  = 0        −λ − 0.142857 = 0, 0 − 0.000043 − λ − λ (−0.000043 − λ ) = 0. Thus the values of λ are − 0.142857 − 0.000043 − λ

λ = 0 and

− 0.000043 − λ = 0

λ = −0.000043.

Third is an analytical solution for MBPJ 2008 based on the equation (13): dS = −2.0510 × 10 − 6 ⋅ I ⋅ S dt dI = 2.0510 × 10 −6 ⋅ I ⋅ S − 0.1429 ⋅ I dt

21

becomes 0 = −2.0510 × 10 −6 ⋅ I ⋅ S , 0 = 2.0510 × 10 −6 ⋅ I ⋅ S − 0.1429 ⋅ I ,  − 0.000002051 ⋅ I and Jacobian matrix, J =   0.000002051 ⋅ I  − 0.000002051 ⋅ S  . 0.000002051 ⋅ S − 0.1429  

We solve value λ at point (0, 0) as follow 0 0  A = J | (0,0) =   0 − 0.1429  ,    then 0 0  1 0 − λ A − λI =   0 − 0.1429  − λ  0 1  = 0        −λ 0 0 = 0, − 0.1429 − λ 0 , − 0.1429 − λ

− λ (−0.1429 − λ ) = 0. Hence the values of λ are

λ = 0 and

− 0.1429 − λ = 0

λ = −0.1429.

We solve values λ at point (69673, 0) as follow  0 − 0.142899  A = J | ( 69673, 0 ) =   0 − 0.00000068 ,    then − 0.142899  0 − 0.142899  1 0 − λ A − λI =   0 − 0.00000068 − λ  0 1  = 0 − 0.00000068 − λ ,       

22

−λ 0

− 0.142899 = 0, − 0.00000068 − λ

− λ (−0.00000068 − λ ) = 0.

Hence the values of λ are

λ = 0 and

− 0.00000068 − λ = 0

λ = −0.00000068.

Lastly is an analytical solution for MPK 2008 based on the equation (14): dS = −1.3769 × 10 − 6 ⋅ I ⋅ S , dt dI = 1.3769 × 10 −6 ⋅ I ⋅ S − 0.1429 ⋅ I , dt

which implies 0 = −1.3769 × 10 −6 ⋅ I ⋅ S , 0 = 1.3769 × 10 −6 ⋅ I ⋅ S − 0.142857 ⋅ I , − 0.00000137688s  − 0.00000137688i  then Jacobian matrix, J =   0.00000137688i 0.00000137688s − 0.142857  .   

We solve at point (0, 0) to get value of λ 0 0  A = J | (0,0) =   0 − 0.142857 ,    0 0  1 0 − λ A − λI =   0 − 0.142857  − λ  0 1  = 0        −λ 0 = 0, 0 − 0.142857 − λ − λ (−0.142857 − λ ) = 0, 0 − 0.142857 − λ ,

23

thus the values of λ are

λ = 0 and

− 0.142857 − λ = 0

λ = −0.142857.

We solve at point (103754, 0) to get values of λ  0 − 0.1428568075  A = J | (103754, 0 ) =   0 0.0000001925 ,     0 − 0.1428568075  1 0  − λ − 0.1428568075  A − λI =   0 0.0000001925  − λ  0 1  = 0 0.0000001925 − λ ,       − λ − 0.1428568075 = 0, 0 0.0000001925 − λ − λ (0.0000001925 − λ ) = 0, thus the values of λ are

λ = 0 and

0.0000001925 − λ = 0

λ = 0.0000001925.
Hence, we summarize the stability points, λ1 and λ2 that have already solved analytically above in Table 10.

24

4.

RESULTS AND DISCUSSIONS

The software applications that we used in this research are Matlab, Maple and Microsoft Office Excel software. The Microsoft Office Excel wills easier us to calculate the data using the formula that we key in since our data is too many. The Matlab software will aid the data obtained by analyzing and graphing the data, while the Maple enables editing and simplification of output obtained which is not only the data but also the equations that we will consider.

4.1

Analysis of Data

Table 10. Summary of equilibrium point and coefficient of Jacobian matrix

Year 2007

Council

Equilibrium Point

λ

1

λ

2

MBPJ

(0, 0) (54875, 0)

0 0 0 0 0 0 0 0

-0.1429 -1.25×10-8 -0.1429 -4.3×10-5 -0.1429 -6.8×10-7 -0.1429 1.925×10-7

MPK

(0, 0) (91575, 0)

2008

MBPJ

(0, 0) (69673, 0)

MPK

(0, 0) (103754, 0)

There is the theorem from Murray (2003) state that to determine either the value of λ is asymptotically stable or not if and only if the value of λ is less than 1 which means − 1 < λ < 1 . But in this case there is no conclusion can be made in the values of λ since all the point is contain the value of zero.

Consequently is a basic reproduction ratio, R0. The mathematical model of R0 is
R0 =

β . Murray (2003) also states that if R0 < 1, then the equilibrium point is locally α

asymptotically stable. As following is the analytical solution of R0 for both municipal and city councils:

25

MBPJ 2007,

R0 =

2.6041 × 10 −6 0.1429 = 1.8223 × 10 −5

MPK 2007,

R0 =

1.5600 × 10 −6 0.1429 = 1.0917 × 10 −5 2.0510 × 10 −6 0.1429

MBPJ 2008,

R0 =

= 1.4353 × 10 −5

MPK 2008,

R0 =

1.3769 × 10 −6 0.1429 = 9.6354 × 10 −6

Since all R0 < 1 , the critical points are stable for all of the council. Now, we do the comparison of the value of R0 between MBPJ 2007 and MBPJ 2008, MPK 2007 and MPK 2008 and also between the two councils in different the years. Begin with Petaling Jaya City Council (MBPJ) shows a decreasing from 2007 to 2008 by 3.87×10-5 which means that the virus has been spreading partly due to decrease urbanization, the population growth not explosive and also partly due to climate change that usually spread of dengue disease happen after the rainy season. On the contrary, the Klang Municipal Council (MPK) also shows a decreasing from 2007 to 2008 by 1.2816×10-6 which means that the urbanization is well planned and organized and also the climate change is certainty. Next is between the year 2007 and 2008. In the year 2007 the difference between MBPJ and MPK is 7.306×10-6, whereas the year 2008 the difference between MBPJ and MPK is 4.7176×10-6. These indicate that in the year 2008 the spread of dengue disease between MBPJ and MPK makes not much difference. However in the year 2007 shows a bit differences that spread of dengue disease in MBPJ more than MPK.

26

In general, we can see that R0 for the MPK 2008 have the smallest ratio compare to other council and year. It also can observe that for the MPK council, the basic reproduction ratio is less than MBPJ council. It may due the action that has been taken by the MPK council to prevent the spread of the dengue disease such as the campaigns in dangerous of dengue disease and the most important is the hygiene factors that have been concerned by MPK council. This action can be followed by MBPJ council, so that the dengue disease can be decreased. For both councils, the current prevention method that should be focused is on eradicating breeding grounds of mosquitoes. The egg-laying places include anything that captures rainwater, such as old tires, pet bowls must be removed and containers for domestic water use should be clean. There also have some encourages from the World Health Organization for use of insecticides together with monitoring to test the vulnerability of mosquito insecticide.

4.2

Analysis of Graph

The solution for the S’, I’, R’ equation is graphed using Matlab software. Below are the graphs of group of susceptible (S’), infected (I’) and recovered (R’) for both councils according year. For following Figure 3, Figure 4, Figure 5 and Figure 6 on the x-axis is time (t) in weeks while on the y-axis is equation S’, I’ and R’. Also on the x-axis the population size rounding up to 10 5 since the value of population is too large and this will make the graph look nice and easy to read.

27

Figure 3. Graph of S’, I’, R’ for MBPJ 2007

Figure 3 is based on equation (11) which is for Petaling Jaya City Council 2007, then in this case on the week of 23 the susceptible group, S’ will approach to zero, while the recovered group, R’ increasing and will approach population size 417030 over time.

Figure 4. Graph of S’, I’, R’ for MPK 2007

Figure 4 above is from equation (12) which is for Klang Municipal Council on 2007 and we can see that on the week of 25 the susceptible group, S’ will reach to zero, while the recovered group, R’ increasing and will approach population size 693956 over time.

28

Figure 5. Graph of S’, I’, R’ for MBPJ 2008

Figure 5 is based on equation (13) which is for Petaling Jaya City Council on 2008, then in this case on the week of 26 the susceptible group, S’ will approach to zero, while the recovered group, R’ increasing and will approach population size 505300 over time.

Figure 6. Graph of S’, I’, R’ for MPK 2008

Figure 6 above is based on equation (14) for Klang Municipal Council on 2008 and we can see that on the week of 29 the S’ will reach to zero, while the recovered group, R’ increasing and will approach population size 725192 over time.

29

Thus, we can see for two differences municipal council in difference year, the graphs obtained the similar pattern of the graph. For all graphs, we can see the relationship between infectious and the host populations are a bit tricky with three-parts at the very least. For all graphs, S represents the proportion of the population who is susceptible to the infectious agent, R represents the recovered population, and I represent the infected population. We can see as time t increase S will decrease and at certain time S will reach zero. Note that the exact shape and inflection of these trend lines will depend on the particulars of the infectious agent. For the I curve it look like the bell-shape but it not completely the symmetric. Note the peak of the I trend will come before the crossover of the recovered and susceptible trends, as it does in this case. As soon as the derivative of the infection rate becomes negative, more people will have recovered than are susceptible and those two trend lines will intersect. Now, we do comparison between the actual data with estimation data to determine whether the both data have the same peak time. It is because we cannot determine the accurate peak time through the graph. For the number of infected persons, I we compare the actual data with estimation data that we take from equation 11, 12, 13 and 14. Since we only consider infected persons to do the comparison, so for equation 11, 12, 13 and 14 we only take I’. Table 11, 12, 13 and 14 below show comparison between actual data and estimation data of I where I is the number of infected persons at time t where t is time in week where Table 11 is for Petaling Jaya City Council (MBPJ) in 2007, Table 12 for Klang Municipal Council in 2007, Table 13 for Petaling Jaya City Council (MBPJ) in 2008 and Table 14 for Klang Municipal Council in 2008.

30

Table 111. Number of infected person of actual and estimation data for MBPJ in 2007 t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 I (actual data) 69 63 85 75 57 58 45 0 15 5 36 35 30 45 30 40 36 31 46 60 45 54 42 47 44 50 34 I (estimation data) 65 59 80 71 54 55 42 0 14 5 34 33 28 42 28 38 34 29 43 56 42 51 40 44 41 47 32 t 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 I (actual data) 37 39 28 25 36 44 50 34 42 48 69 123 50 85 39 56 0 58 56 47 38 29 0 0 0 0 0 I (estimation data) 35 37 26 23 34 41 47 32 39 45 65 115 47 80 37 53 0 54 53 44 36 27 0 0 0 0 0

As we can see from Table 11 above, the number of infected person from the actual data is peak at week 39 with the infected person is 123 persons meanwhile for the estimation data, the number of infected person are peak at week 39 with the infected

31

person is 115 persons. Both actual data and estimation data are peak at the same week.
Table 112. Number of infected person of actual and estimation data for MPK in 2007 t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 I (actual data) 9 6 8 7 12 4 4 6 4 0 6 7 7 3 0 4 4 6 7 17 10 4 4 6 8 7 11 I (estimation data) 12 8 10 9 15 5 5 8 5 0 8 9 9 4 0 5 5 8 9 22 13 5 5 8 10 9 14 t 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 I (actual data) 11 0 6 5 6 10 1 10 2 3 0 3 4 5 2 3 0 4 4 0 10 6 0 0 0 0 0 I (estimation data) 14 0 8 6 8 13 1 13 3 4 0 4 5 6 3 4 0 5 5 0 13 8 0 0 0 0 0

As we can see from Table 12 above, the number of infected person from the actual data is peak at the same week with the estimation data which are at the week 20 where

32

the infected person for the actual data is 17 persons and for the estimation data is 22 persons infected. Table 13. Number of infected person of actual and estimation data for MBPJ in 2008 t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 I (actual data) 18 15 20 19 15 22 23 33 34 32 23 23 31 37 25 23 22 33 38 45 35 34 24 29 35 32 46 I (estimation data) 16 13 18 17 13 20 21 29 30 29 21 21 28 33 22 21 20 29 34 40 31 30 21 26 31 29 41 t 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 I (actual data) 38 26 37 26 43 28 26 22 22 20 24 24 12 25 24 14 16 15 20 23 28 34 24 28 27 37 0 I (estimation data) 34 23 33 23 38 25 23 20 20 18 21 21 11 22 21 12 14 13 18 20 25 30 21 25 24 33 0

From the Table 13 above, the peak time for actual data is at week 27 with 46 persons has been infected. For the estimation data, the peak time is at week 27 too but number

33

of persons who is infected is 41. So we can see for both data the peak time is alike which at the week 27 with the number of infected people for estimation data are 5 persons less than actual data. Table 14. Number of infected person of actual and estimation data for MPK in 2008 t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 I (actual data) 80 71 61 69 58 53 65 50 42 53 36 45 46 61 63 60 79 75 69 79 62 65 63 71 62 74 74 I (estimation data) 94 83 71 81 68 62 76 59 49 62 42 53 54 71 74 70 92 88 81 92 72 76 74 83 72 86 86 t 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 I (actual data) 68 51 52 52 52 64 57 51 42 45 63 50 39 48 49 32 40 35 56 47 47 71 54 90 77 50 0 I (estimation data) 79 60 61 61 61 75 67 60 49 53 74 58 46 56 57 37 47 41 65 55 55 83 63 105 90 58 0

34

From the Table 14 above, the peak time for actual data is at week 51 with 90 persons has been infected. For the estimation data, the peak time is at week 51 too, but number of persons who is infected is 105. So we can see for both data the peak time is alike which at the week 51with the number of infected persons for estimation data is 15 people over than actual data. We can conclude that α and β obtain from the SIR model are useful to estimate the spread of dengue disease for the following year, since the peak time for the actual data and the estimation data are same. We can make the estimation trend of the spread of dengue disease and take the early action to prevent it.

5.

CONCLUSION

The using SIR model of dengue disease in city and municipal council of Selangor are important especially for the high-rise dengue disease cases with it at urban area to ensure all area safe from dengue disease and reducing the number of people infected. From the stability of equilibrium point and reproduction ratio, we can see the trend of the susceptible, infection and recovered population for the city and municipal council area are asymptotically stable. We can conclude that the infected person in the population increase or decrease is depending on the prevention that has been taken by each city and municipal council. The spread of the dengue disease in the MPK is under control rather than MBPJ. It means the action taken by the MPK is better and useful compare to the MBPJ were the recovered population increase within the time.

6. RECOMMENDATIONS AND FUTURE STUDY

In the future study, this research can be expanded by using other method such as Multistage Adomian Decomposition Method (MADM), Runge-Kutta, Fourier series

35

and other ordinary differentiate equation to determine the parameter estimation of SIR model other than using the Euler Method. From the equation that obtain, the free equilibrium point or stability point can be interpret from the graph of the simulation from the MATLAB. Another recommendation is to make comparison within urban area and rural area to see the different between each area with the small population and also make consideration on the birth rate and death rate.

36

7.

REFERENCES

Bianca, N. (2009). Infecting mosquito may keep them from infecting us; lifeshortening bacterium could beat mosquito-borne disease, Scientific American. Retrieved 28 December 2012 from http://www.scientificamerican.com/article.cfm?id=infecting-mosquitoes Boelle, P.Y., Bernillon, P., Desenclos, J.C. (2009). A preliminary estimation of the reproduction ratio for new influenza a(H1N1) from the outbreak in Mexico, March-April 2009, Eurosurveillance 14 (19) Derouich, M., Boutayeb, A., & Twizell, E.H. (2003). Research a model of dengue fever, BioMedical Engineering OnLine 2003. Johnson, T. (2009). Mathematical modeling of diseases:susceptible-infectedrecovery (SIR) model. Murray, J.D. (2003). Nonlinear difference equations. (3th ed.) and biomedical applications. Mathematical Biology, 69-73. . Spatial models

Nuraini, Y., Harun, B., & Salemah, I. (2010). Parameter estimation of the sir model using The Multistage Adomian Decomposition Method (MADM), Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA2010), University of Tunku Abdul Rahman, Kuala Lumpur, Malaysia. Pongsumpun, P., & Kongnuy, R. (2007). Model for the transmission of dengue disease in pregnant and non-pregnant patients, International Journal of Mathematical models and methods in Applied Sciences. Vol. 1, No. 3. Pongsumpun, P., & Tang, I.M. (2001). A realistic age structured transmission model for dengue hemorrhagic fever in thailand. Vol. 32, No. 2. World Health Organization. (1999). Dengue hemorrhagic fever: Diagnosis, treatment, preventation and control (2nd ed.).

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