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Reduction of mosquito populations will, at least, reduce substantially the transmission of malaria disease. One potential method of achieving this reduction is the environmentally-friendly population control method known as the Sterile Insect Control (SIT) method. The SIT method has so far not been widely used against insect disease vectors, such as mosquitoes, because of various practical difficulties in rearing, sterilization and distribution of the parasite population. For mosquitoes, male-only release is considered essential since sterile females will bite and so may transmit disease, whereas male mosquitoes do not bite. This work concerns the mathematical modelling of the effectiveness of Sterile Insect Technique for Aedes aegypti mosquitoes, when the female sexual preference is incorporated. We found that for a released value of the sterile male mosquito below 40,000, the wild mosquito population decreases over time while the sterile male mosquito population increases. Therefore, the transmission of malaria and dengue infection declines because the sterile male mosquitoes dominated the environment. We also found that for a released value of the sterile male mosquito above 40,000, the wild mosquito population decreases and the sterile male mosquito population decreases as well. Therefore, if the injection of sterile male mosquitoes is large enough, the environment will be rid of mosquitoes over time. The result also shows that if sexual selection is incorporated into a reaction diffusion system, modelling the spread of Aedes aegypti mosquitoes, the Sterile Insect Technique (SIT) will still be a successful control measure.

Malaria is a major life-threatening vector-borne disease transmitted through mosquitoes. The disease got its name from bad air (malaria) as it was thought that the disease came from fetid marshes. Later in 1880, it was discovered that the real cause of malaria was Plasmodium [

The disease is more prevalent in the tropical and sub-tropical regions of the world and causes more than 300 million acute illnesses and at least one million deaths annually [

In vector borne disease control strategy, it is important to include the mating behavior of the mosquito, which is an aspect of mosquito biology that is not fully understood [

The Sterile Insect Technology has been tried in a number of scenarios, as an attempt to combat malaria and dengue with little success. Many of the earlier approaches were based on ordinary differential equations. They made several assumptions such as 1:1 sex ratio [

The objective of this study was to develop a computer program for solving the model equations developed by [

PDE model for Aedes aegypti mosquito incorporating female sexual preference is proposed by Parshad and Agusto (2011).

This model is an extension of the model in [

The per capita mating rates of an unmating female with a wild male mosquito are given by:

and

where:

The per capita mating rate of a female with a sterile male is given by:

where:

The model is described in Equation (4) and the description of the variables and parameters of the model are shown in

On

Assumptions made by Parshad and Agusto (2011):

· A female mosquito mates once in its life, and oviposit its eggs in different places during its entire life [

· The population of females with sexual preference is maintained by mutation from female without sexual preference at the rate

· The females with preference do not mate with the sterile male (even in small probability). This is to incorporate selection in the model, and helps to clearly differentiate between the classes I and P. Note, there is no conclusive evidence, that sexual selection exists in mosquito mating, however there are a number of studies, [

· We assume what is known as a mutation-selection balance, [

In simulating, model Equation (4), the parameter values in

Variable | Description |
---|---|

Immature phase of insect | |

Single females with male preference | |

Single females with wild male preference | |

Mating fertilized females | |

Wild males | |

Sterile males | |

Parameter | Description |

Carrying capacity related to the amount of available nutrients and space | |

Intrinsic oviposition rate | |

Mating rate of natural insects | |

Mating rate if SIT | |

Percentage of reduction of mating capacity | |

Ability of dispersion | |

Mature rate to adulthood | |

Female mature rate to adulthood | |

Male mature rate to adulthood | |

Natural mortality rate | |

Release rates of sterile male mosquitoes |

Courtesy: [

0.50 | 0.50 | 0.50 | 0.1 | 0.1 | |

5.0 | 0.075 | 0.5 | 1.0 | 600 | 0.7 |

2000 | 1000 | 2500 | 1500 | 2000 | 1200 |

The computer program for the simulation is presented in Appendix 1.

In the simulation presented on Figures 2-5, we used model Equation (4), the parameter values in

The sterile insect technique (SIT) is a biological control which disorders the natural reproductive process of insects; male insects are first made sterile by gamma radiation before releasing them in large numbers into the environment to mate with the native wild insects. Insect populations can be controlled by the release of large numbers of sterile males. Thus, if a female mate with a male that has no sperm or whose sperm was rendered unviable, this female will have fewer or no offspring. When many sterile males are released, the local population tends to decline or become wiped out. For population control, the crucial parameter is the ratio of the number of released sterile males to the number of males in the local population, which ideally should be around 10:1 [

As demonstrated on

toes do not bite. These released insects compete for mates with wild males; a wild female mating with a released sterile male has no or fewer progeny, so the population tends to decline. This means that the sterile male mosquitoes have dominated the environment. Hence if the mosquito population has not grown yet sufficiently or it has been reduced by some other measure e.g. destroying the breeding places, it can be controlled by release of sterile males.

As shown in

This implies that if sufficient sterile male mosquitoes are released for a sufficient period, the wild mosquito population decreases to zero and so the environment is dominated by the sterile male mosquitoes and with time these sterile male mosquitoes start dying since there is no reproduction. Therefore, if the injection of sterile male mosquitoes is large enough the environment will be eradicated completely of mosquitoes over time.

Essentially our results show that if sexual selection is incorporated into a reaction diffusion system, modeling the spread of Aedes aegypti mosquitoes, the sterile insect technique can still be a successful control measure, if the injection of sterile males is large enough.

From the simulation, it was revealed that the success of SIT depends on the parameters of the wild mosquitoes as well as on parameters related to the sterile male mosquitoes.

We established that for a released value of the sterile male mosquito below 40,000, there was reduction in the wild mosquito population and domination of the sterile male mosquitoes in the environment. For a released value of the sterile male mosquito above 40,000, there was eradication of mosquitoes in the environment over time as the sterile male mosquito population decreases. The result shows that if sexual selection is incorporated into a reaction diffusion system, modelling the spread of Aedes aegypti mosquitoes, the sterile insect technology can still be a successful control measure if the injection of sterile males is large enough. Therefore, if sufficient sterile insects are released for a sufficient period, the target population will be controlled or even locally eradicated.

This project was supported by Lagos State Government (Local Scholarship Award). The authors would like to thank Mogbojuri Babatunde (Lead software developer/system engineer) at Global Accelerex Ltd. who was there in the course of this project to provide answers to every professional question.

The computer Program for the simulation of a PDE model developed by Parshad and Agusto 2011 for Aedes aegypti mosquito using SIT when the female sexual preference is incorporated. Function mc = mosquito control

This section of the code declares all constants

mu = 0.5; % mortality rates of the unmating female with and without sexual preference

muf = 0.5; % mortality rates of the fertilized female with and without sexual preference

mum = 0.1; % mortality rates of the wild males

mums = 0.1; % mortality rates of the sterile males

Bi = 1.0;

del = 0.12;

Bp = 1.0;

Bs = 1.0;

phi = 50; % oviposition rate per female mosquito

gamma = 0.1; % the rate at which the aquatic population becomes winged mosquitoes

r = 0.5; % proportion that transforms into female

C = 600; % carrying capacity effect

alpha = 4000; %rate at which sterile males are released and sprayed

This section of the code solves the system of PDE’s

% the following definitions apply

This section allows user to specify initial value of variables for the system of PDE’s

Ai0 = input (“Enter a value for Ai (0) = ”);

Ap0 = input (“Enter a value for Ap (0) = ”);

I0 = input (“Enter a value for I (0) = ”);

P0 = input (“Enter a value for P (0) = ”);

Fi0 = input (“Enter a value for Fi (0) = ”);

Fp0 = input (“Enter a value for Fp (0) = ”);

M0 = input (“Enter a value for M (0) = ”);

Ms0 = input (“Enter a value for Ms(0) = ”);

This section solves the system of PDE’s

This section generates the graph of the system of PDE's

Plot (t,x(:,1),t,x(:,2),t,x(:,3),t,x(:,4),t,x(:,5),t,x(:,6),t,x(:,7),t,x(:,8));

xlabel ('time (days)');

ylabel (“population”);

legend ('Ai', 'Ap', 'I', 'P', 'Fi', 'Fp', 'M', 'Ms')

title (“Graph showing population distribution of mosquitoes”);

figure;

plot (t,x(:,1),t,x(:,2),t,x(:,3),t,x(:,4),t,x(:,5),t,x(:,6),t,x(:,7));

xlabel (“time (days)”);

ylabel (“population”);

legend ('Ai','Ap','I','P','Fi','Fp','M')

title (“Graph showing population distribution of mosquitoes”);

figure;

plot (t,x(:,8));

xlabel (“time (days)”);

ylabel (“population”);

legend (“Ms”)

figure;

plot (t,x(:,7));

xlabel (“time (days)”);

ylabel (“population”);

legend (“M”)

figure;

plot (t,x(:,6));

xlabel (“time (days)”);

ylabel (“population”);

legend (“Fp”)

figure;

plot (t,x(:,5));

xlabel (“time (days)”);

ylabel (“population”);

legend (“Fi”)

figure;

plot (t,x(:,4));

xlabel (“time (days)”);

ylabel (“population”);

legend (“P”)

figure;

plot (t,x(:,3));

xlabel (“time (days)”);

ylabel (“population”);

legend (“I”)

figure;

plot (t,x(:,2));

xlabel (“time (days)”);

ylabel (“population”);

legend (“Ap”)

figure;

plot (t,x(:,1));

xlabel (“time (days)”);

ylabel (“population”);

legend (“Ai”)

Algorithm for Mosquito control model

1) Define model parameters and variables

2) Input values of constants_{, , },

3) Specify model equation

4) Input initial values of variables

5) Determine solution method for system of equations (4th order Runge-Kutta method)

6) Specify step increment and simulation range for independent variable (time)

7) Solve model equations using solution method specified above

8) Dimension array and align to grid

9) Start iterations

10) Compute variables at each point

11) Continue loop until last point

12) Print result

13) Plot result and print graphs

14) End simulation