




































 

 

 
1 

© 2018 Conscientia Beam. All Rights Reserved. 

THE SIMULATION OF EGG HATCHING TREND OF AEDES AEGYPTI ASSOCIATED 
WITH RAINFALL DISTRIBUTION   

 

 

 Najir Tokachil1+ 

 Nuraini Yusoff2 

 

1Faculty of Computer and Mathematical Sciences Universiti Teknologi 
MARA Cawangan Negeri Sembilan Kampus Seremban, 70300, Negeri 
Sembilan, Malaysia 

 
2Faculty of Computer and Mathematical Sciences Universiti Teknologi 
MARA, 40450 Shah Alam, Malaysia 

 
 

 
(+ Corresponding author) 

 ABSTRACT 
 
Article History 
Received: 4 April 2018 
Revised: 15 May 2018 
Accepted: 18 May 2018 
Published: 21 May 2018 
 
 

Keywords 
Aedes aegypti,  
Lefkovitch matrix 
Dengue,Stage-Structured 
Population Dynamics 

 
Aedes aegypti is primary vector of Zika virus, dengue, and chikungunya which go 
through four different stages of life cycle: Egg, Larva, Pupa, and Adult. The 
mosquito might be able to lay 100 eggs at a time. The eggs can hatch when there is 
water and submerged in the water, but it can survive without water almost to 8 
months by sticking to the walls of a container. In this research, the trend of egg 
hatching was simulated by using a stage-structured Lefkovitch matrix model. The 
matrix model consists of a function of egg hatching which associated and depends 
on rainfall distribution. The purpose of the function is to identify the relationship 
between egg hatching and rainfall which provide water to egg. Using the same 
mathematical model, the population of adult Aedes aegypti has also simulated which 
the population related to egg simulation and rainfall. The graphical results were 
shown to illustrate the relationship between the rainfall, egg number and adult 
Aedes aegypti. We found that there was a significant relationship between the 
rainfall and these two stages of mosquito life cycle which are egg hatching number 
and adult Aedes aegypti. However, it is recommended to look for more possible 
factors such as the size of an opening container and humidity which possible to 
contribute to the development of mosquito to acquire more accurate results. 

 

Contribution/Originality: The study contributes in the existing literature which stated that both factor rainfall 

and temperature involved in the development of mosquito life cycle. We found that, the rainfall distribution factor is 

the most significant to identify the trend of development of mosquitos’ population and meanwhile the factor of 

temperature influences more on the increase of dengue virus transmission.  

 

1. INTRODUCTION 

Dengue fever is a contagious tropical or a mosquito-borne viral disease which transmitted by female Aedes 

aegypti mosquito. The dengue can be spread into two forms which are Dengue Fever (DF) and Dengue 

Hemorrhagic Fever (DHF). Patients with DF might go to develop DHF which can evolve to severe form or death 

known as Dengue Shock Syndrome (DSS) [1]. DHF and DSS can occur in both children and adults. 

The Aedes aegypti originates from Africa and now exist globally in tropical and subtropical region. The 

distribution of this mosquito might through global trade and shipping activities. These mosquitoes usually breed 

around the human habitat and bite during the day. They are easily recognizable by their peculiar white spotted 

body and legs. The infected female Aedes aegypti can spread dengue virus mainly through the bites [2]. The 

transmission occurs when the mosquito bites a person who already gets infected by dengue fever and the virus will 

Animal Review 
2018 Vol. 5, No. 1, pp. 1-7 
ISSN(e): 2409-6490 
ISSN(p): 2412-3382 
DOI: 10.18488/journal.ar.2018.51.1.7 
© 2018 Conscientia Beam. All Rights Reserved. 

 
 
 

 
 

 
 

https://orcid.org/0000-0002-3378-8805
https://orcid.org/0000-0003-3675-8019
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Animal Review, 2018, 5(1): 1-7 

 

 
2 

© 2018 Conscientia Beam. All Rights Reserved. 

enter the mosquito. The mosquito bites a susceptible person, and then dengue virus will spread in that person [3]. 

Besides, the infected female mosquitos also can spread the virus to their eggs [4].  To produce their eggs, the 

female mosquitoes require a protein which can be obtained from human blood. However, adult male mosquitoes feed 

on nectar, and they do not involve both transmitting virus and reproducing. 

There are four main stages of the life cycle which are eggs, larvae, pupae and adult. Many articles have stated 

that the factors of temperature and rainfall contribute to the increase of number of mosquito which the main vector 

of spreading dengue virus [5]. However, in Malaysia the temperature was recorded as almost unchanged compared 

than rainfall distribution. Therefore, the purpose of this study is to simulate the first stage of mosquito which is egg 

stage associated with rainfall distribution using Lefkovitch matrix model. Using the same mathematical model, the 

population of adult Aedes aegypti was also simulated because the adult is the main life stage of mosquito which 

involved in both transmitting virus and reproducing egg. A model proposed in previous research [6]; [7]; [8] will 

be considered and discussed in order to achieve the objectives of this research. We focused on Shah Alam city since 

the reported cases of dengue are always high and only first three months in 2016 was considered because these 

months have high dengue cases compared to others period. 

 

2. METHOD 

The projection matrix of Aedes aegypti life cycle was considered to forecast the population of mosquito based on 

distribution of rainfall. In this section, all the methods used to achieve the objectives of this research will be clearly 

explained. Generally, the population model is described in matrix algebra with consist of transaction rate of Aedes 

aegypti life cycle.  

 

2.1. Population Projection Matrix 

In this research, a Lefkovitch matrix model was used to simulate each stage in the life cycle of Aedes aegypti 

mosquitoes. The mosquitoes have gone through four life cycle stages: egg, larva, pupa, and adult. However, in this 

research, the adult stages are divided into two stages namely Adult 1 and Adult 2. The Adult 1 was described as a 

mosquito that can produce and lay the first eggs to convert into Adult 2.  The stage of Adult 2 is an adult has been 

laying egg more than one time and gone through several egg laying processes. The Lefkovitch matrix model was 

used to simulate the population number of Aedes aegypti because it is one of the best stage-structured matrix models 

[9]. Only female mosquitoes are considered because they directly involved in egg laying process. 

 

2.2. Lefkovitch Matrix Model 

The population sizes with respect to age are difficult to determine precisely rather than considering the stages 

of population life cycle. Therefore, the stage-structured matrix model was constructed by considering the growth of 

mosquito life cycle stages and breeding maturity [10]. The population of Aedes aegypti mosquito is easier to identify 

based on their life cycle than its age because each life stage can be differently classified based on the growth of 

physical size. In this section, the simulation number of mosquito population at any time t will be assigned 

as ( )N t and the transition matrix A is the transformation based on mosquito life cycle. The simulation of population 

at time 1t  can calculate by multiply matrix A and ( )N t .  

                                                  

 
 
 
   
 
 
 
 

1 4 5

1 2

2 3

3 4

4 5

0 0

( ) 0 0 0

0 0 0( 1) ( ) (1)

0 0 0

0 0 0

P F F

E t P

G PN t N t

G P

G P

                                                                      



Animal Review, 2018, 5(1): 1-7 

 

 
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© 2018 Conscientia Beam. All Rights Reserved. 

The transition matrix A consist of five elements which completely depending on stage of mosquito life cycle, i  

from egg, larva, pupa, Adult 1 and Adult 2. The element 
i

P is referred as the proposition of Aedes aegypti survive and 

stay in the same stage i . The 
i

G  is the proposition of surviving and growing of the mosquito from stage i  into the 

next stage ( 1)i   . The parameter 
i

F  is described as the fertility of mosquito in stage i . The fertility rates are set 

only for adult stages which the mating process and producing eggs occurred during this stage.  

 

2.3. Transition Rate of Aedes Aegypti Life Cycle 

We let that the survival rate and the duration rate for stage i respectively denoted as 
i

s and 
i

d . The transition 

rate, 
i

P  can be computed using the formula as follows. 

                                                                    






1(1 )
(2)

1

i

i

d

i i

i d

i

s s
P

s
 

Since the plus of proportion, 
i

G  and 
i

P  is equal to the total population of stage i survive, then transition rate, 

i
G  can be obtained by using the formula as follows. 

                                                                






(1 )
(3)

1

i

i

d

i i

i d

i

s s
G

s
 

The fecundity rate of F2 and F3 are equal to 0 since only adult 1 and adult 2 could produce eggs. Therefore, only 

the transaction rate
4

F and
5

F are considered for the transaction matrix A . Since, the time duration required by each 

stage to survive and grow might be different then the time estimation must be assumed based on environmental and 

biological factors. 

 

2.4. Transition Rate of Egg Hatching 

Water is the main source in the process of egg hatching. Therefore, we assumed that only uncovered and 

opening containers can be filled with amount of water caused by rainfall. The amount of water stagnated in the 

container will be a breeding site for the mosquito. In this situation, we considered that the heavy rainfall 

distribution possibly influences the egg hatching process by overflowing the water in containers. The Egg hatching 

process was directly associated with rainfall distribution can be described as a function of water depth with respect 

to time t (days). 

                                                              
 

  
 

2

1 1 1

max

( 1)
( ) (4)

c

wd t
E t s c

wd
 

The parameter 1s  is the proportion of the Aedes aegypti in egg stage will survive and move to the larva stage, 

 1twd  was taken as the amount of rainfall or stagnant water in a container at time t disregarding the shape of 

the container. The maxwd is the average depth of the container. Meanwhile, 1c  is probability of egg hatching and 

2c is the pattern of egg hatching [6];[7];[8]. 

 

2.5. Simulation of Egg Hatching and Adult Stage Associated with Rainfall 

The proportion of iG ,
i
P and  1

E t  can be calculated by using the information in Table 1 obtained from our 

previous studies(referred [7];[8]). 

 

 

 

 



Animal Review, 2018, 5(1): 1-7 

 

 
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© 2018 Conscientia Beam. All Rights Reserved. 

Table-1. The Survival Rate and Duration of Each Stage in Mosquito Life Cycle 

i  Stage Survival rate, 
i

s  Age (days),
i

d  

1 Egg 0.9890 < 4 
2 Larvae 0.9898 4 – 8 
3 Pupae 0.9898 9 – 10 
4 Adult 1 0.9100 11 – 14 
5 Adult 2 0.9100 15 - 24 

 

 

The daily rainfall of January, February and March in year 2016 were considered because the monthly cases of 

Dengue fever are reported higher than other months in the city of Shah Alam. The data of rainfall obtained from 

Meteorology Department, Malaysia. The measurement of rainfall is in millimeter (mm). 

 

2.6. Implementation 

In this section, the simulation is obtained where the matrix A  is multiplied by the initial population of 

mosquito, ( )N t . We set the initial number of eggs in the container is 1000 eggs. The information in Table 1 will be 

used to obtain each rate in matrix A  using the formula 
i

G ,
i

P and  1
E t . The values of  1

E t will change depend on 

Rainfall distribution. The value of 
1

c and 
2

c are 0.27901 and 5 respectively. 

   
   
   
    
   
   
   
   

1

0.74622 0 0 8 16 1000

(1) 0.79588 0 0 0 0

0 0.19392 0.49744 0 0 0(1)

0 0 0.49240 0.71360 0 0

0 0 0 0.19640 0.85260 0

E

N  

The simulation of mosquito population for the following days, ( 1)N t   were obtained by using Maple where 

the population of each stage in the mosquito life cycle change exactly depend on daily rainfall distribution. 

 

3. RESULT AND DISCUSSION 

The simulation of the mosquito population obtained from model, ( )N t  are illustrated graphically into Figure 1, 

Figure 2 and Figure 3 to observe the trend between simulation of egg number associated with rainfall and the 

influences on the number of adult population. In the previous section, we mentioned that month January, February 

and March in 2016 were considering since the dengue cases were reported high in these months. The observation 

on the trend of egg number is important because it influences the adult population. The further explanations will 

discuss in this section. In Figure 1, we found that the trend of egg number during the month was very significant 

starting in the week 10. The rainfall received consecutively between week 11 to 18 allowing adult mosquitoes to 

produce eggs because the rainfall might provide the breeding sites [8]. Although, the frequency of receiving rain in 

the next week is low but the number of eggs continue to increase. This could be occurred due to receiving rain 

between the week 11 to week 18 provided enough breeding site for adult mosquito to lay their eggs. This situation, 

it also affects the number of adult mosquitoes that increase gradually based on the number of eggs. 



Animal Review, 2018, 5(1): 1-7 

 

 
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© 2018 Conscientia Beam. All Rights Reserved. 

 
Figure-1. The Relationship of Egg Number, Adult and Rainfall Distribution on January 2016 

 

At the beginning of February (refer Figure 2), the number of eggs was high but the number goes down suddenly 

starting from week 8 to 25. We noticed that the frequency of receiving rainfall in this month is less than January. 

We could say that the rainfall distribution received in this month might affect the existence of mosquitoes breeding 

site where there might be none and lack of stagnant water in opening containers. However, this number increased 

drastically starting from the week 25 onwards. Hence, the adult mosquitoes also have the same trends as eggs. 

 
Figure-2. The Relationship of Egg Number, Adult and Rainfall Distribution on February 2016 

 

Referring to Figure 3, the number of eggs keeps increasing from the last month. This situation persists until 

week 10 in March and suddenly, the trend of eggs number goes down in the following weeks. We can observe there 

are five consecutive weeks received high rainfall starting from week 10 until week 14. The high rainfall might cause 

the stagnant water in the containers overflow and if there are eggs in the container, the abundant water will wash 

out the eggs from the container. The egg number will decrease, and the process of egg hatching almost does not 

occur. This situation also influences the development of adult mosquitoes and production of egg as well [11]. The 

fluctuation of egg number trend continued to decline until the end of the month which throughout the weeks the 

receiving rains were very low, and the weather become dry. However, the adult mosquitos that have arisen from the 

weeks before are very active because this period is most appropriate for mosquitoes to spread the virus to humans. 

It is advisable to keep people cautious if this trend of rain happen. Eliminate mosquito breeding site is one of the 

best methods that can help reduce the development of mosquitoes. 



Animal Review, 2018, 5(1): 1-7 

 

 
6 

© 2018 Conscientia Beam. All Rights Reserved. 

 
Figure-3. The Relationship of Egg Number and Rainfall Distribution on March 2016 

 

4. CONCLUSION 

Using the stage-structured matrix model for mosquito simulations gives some insight into the development of 

the life cycle of mosquitoes. This simulation is an alternative method to reduce dengue disease by studying the life 

cycle of mosquitoes. In this study, the factor of rainfall is the main role because it is seen that rain can provide 

breeding sites for the mosquitoes. Based on the results we get from this study, we found that the number of eggs 

will increase if there is a moderate rainfall distribution that provides enough water in the container. However, high 

rainfall can affect the amount of eggs which the heavy rains cause water in containers to overflow and carry eggs 

out from the water and containers. This situation causes the hatching process to almost do not occur and then 

affects the number of adult mosquitoes. The spread of dengue virus also decreases but the dry or moderate weather 

will make the adult mosquito active in the spread of the dengue virus. However, future studies are recommended to 

consider other factors such as humidity and mosquito habitat areas. 

 

Funding: This research was supported by Universiti Teknologi MARA (UiTM) under Geran Dana 
Pembudayaan Penyelidikan (iRAGS),600-RMI/DANA 5/3/IRAGS (6/2015). 
Competing Interests: The authors declare that they have no competing interests.  
Contributors/Acknowledgement: The author(s) would like to thank to Institute of Research Management 
and Innovation (IRMI) UiTM as a management center of this grant. 

 

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http://dimacs.rutgers.edu/
https://scholar.google.com/scholar?hl=en&q=An%20analysis%20of%20life%20history%20evolution%20in%20terms%20of%20the%20density-dependent%20lefkovitch%20matrix%20model
http://dx.doi.org/10.1016/0025-5564(92)90091-a

