_{1}

^{*}

The anthropophilic and peridomestic female Aedes aegypti bites humans to suck blood to maturate fertilized eggs, which are laid in appropriate recipients (breeding sites). These eggs can hatch in contact with water releasing larvae, or can be stored in a dormant state (quiescence), which last for extended periods. Taking into account this ability of eggs of A. aegypti mosquitoes, mathematical model is developed taking into account four successive quiescence stages. The analysis of the model shows that the ability of the eggs surviving in dormant state in adverse abiotic conditions, depending on the model parameters, can increase the fitness of mosquito population; in other words, the capacity of the mosquitoes generating offsprings is increased.

The population dynamics of mosquitoes Aedes aegypti is clearly dependent on abiotic factors, with serious implications for dengue transmission. By using estimated entomological parameters dependent on temperature, including the dependency of these parameters on rainfall, the seasonally varying population size of mosquito A. aegypti was evaluated by a mathematical model [

The eggs of the mosquito A. aegypti possess the ability to undergo an extended quiescence hosting a fully developed 1^{st} instar larvae within the chorion, and this life history traitpharate larvae can withstand months of quiescence inside the egg where they depend on stored maternal reserves. Therefore, the duration of quiescence and extent of nutritional depletion may affect the physiology and survival of larvae that hatch in a suboptimal habitat [

Silva and Silva [

The paper is structured as follows. In Section 2, a model for A. aegypti population is formulated encompassing quiescence eggs, and in Section 3 the model is analyzed, determining the equilibrium points, and performing the stability analysis of these points. Section 4 presents discussion, and conclusion is given in Section 5.

Embryonic development of the eggs of the mosquito A. aegypti is completed approximately within 3 days after oviposition, and a fully developed 1^{st} instar larva resides within the chorion of the egg in a dormant state referred to as quiescence. Pharate 1^{st} instar quiescent larvae will hatch out immediately upon exposure to the appropriate stimulus; in this way quiescence differs from diapause, which is a hormonally controlled and pre-programmed state of developmental arrest in which the larvae are refractory to hatching stimuli for an extended period of time. As a result of this life history trait, A. aegypti produce eggs that, in addition to being desiccation resistant, can withstand months of dormancy depending on stored maternal reserves [

The life cycle of A. aegypti encompasses an aquatic phase (egg, larva and pupa) followed by winged (adult) form. In Yang et al. [

In the modelling, it is assumed that all eggs laid by mosquitoes enter in the first quiescence stage, which number at time t is designated as

it is assumed that the eggs in quiescence stage

The above descriptions are also valid for the quiescence eggs

Based on the foregoing descriptions of model parameters and variables, the dynamics of mosquito population encompassing quiescence eggs is described by the system of differential Equations

This system of equations is analyzed in the steady state. A simplified version of this modelling is given in Appendix.

The system of Equations (1) is dealt with determining the equilibrium points, and assessing the stability of these points.

Before determining the equilibrium points, let the following parameters be defined. The quiescence eggs at stage i can go to next quiescence stage

for

for

for

There are two equilibrium points. The first equilibrium is the absence of mosquito population, designated by

which is referred to the trivial equilibrium point.

The second equilibrium is the mosquito population being well established in a region (or community), the non-trivial equilibrium

The coordinates of the non-trivial equilibrium are written in terms of the previously defined parameters, which are

and the number of adult mosquitoes

where

with

which is the overall production of larvae by all compartments of eggs. Clearly, the non-trivial equilibrium point is biologically feasible if

Let

The overall production of eggs

The stability analysis of the equilibrium points

The Jacobian matrix evaluated at the equilibrium point

where the

with

with

with

which can be written as

a polynomial of

The independent term of the characteristic Equation, designated by L_{0}, which is given by

where

For the trivial equilibrium

Hence, according to Leite et al. [

The results obtained in previous section are dealt with to assess the influence of the quiescence eggs in the size of A. aegypti population. First, the study is done qualitatively, and, then, the results are confronted with experimental data.

The basic offspring number

Notice that this is obtained by letting

Let first be defined the productivity of each hatchable stage

number of died eggs is

The difference between

where

and

where

with

Notice that the parameters

There are four hatchable states

A.

In this case,

B.

In this case,

B.1.

In this case,

Hence, the quiescence eggs increase the fitness of mosquito population

B.2.

In this case,

Hence, the quiescence eggs increase the fitness of mosquito population

C.

In this case,

Hence, the quiescence eggs increase the fitness of mosquito population

D.

In this case,

Hence, the quiescence eggs increase the fitness of mosquito population

Summarizing, the quiescence eggs increase the fitness of mosquito population if one of hatchable stages from_{1}. Besides the productivity indexes

The transition parameters

Hence, depending on the productivity index

The theoretical results obtained in foregoing section are compared with the results of the experiments carried out by Silva and Silva [

Experiment number | Quiescence (days) | Number of eggs | Eclosion (eggs × days^{−}^{1}) | Eclosion (%) |
---|---|---|---|---|

1 | 3 | 807 | 86.1 | 85.4 |

2 | 32 | 698 | 5.3 | 41.1 |

3 | 63 | 586 | 6.4 | 36.0 |

4 | 91 | 738 | 12.1 | 47.7 |

5 | 121 | 749 | 13.2 | 97.2 |

6 | 154 | 800 | 1.6 | 1.3 |

7 | 273 | 612 | 8.6 | 4.3 |

8 | 337 | 611 | 1.0 | 0.3 |

9 | 427 | 842 | 5.6 | 10.9 |

10 | 462 | 800 | 1.0 | 0.5 |

11 | 492 | 1708 | 1.0 | 0.2 |

Based on

Observing the last column of

The estimated eclosion

Experiment number | Per-capita eclosion rate (days^{−}^{1}) | Per-capita mortality rate (days^{−}^{1}) |
---|---|---|

1 | 0.1067 | 0.0182 |

2 | 0.007593 | 0.0109 |

3 | 0.01092 | 0.0194 |

4 | 0.01640 | 0.0180 |

5 | 0.01762 | 0.00051 |

6 | 0.002 | 0.1518 |

7 | 0.01405 | 0.3127 |

8 | 0.00164 | 0.5439 |

9 | 0.00665 | 0.05437 |

10 | 0.00125 | 0.2488 |

11 | 0.000585 | 0.2922 |

Stage—i | σ_{i} (days^{−}^{1}) | μ_{i} (days^{−}^{1}) | p_{i} | α_{i} (days^{−}^{1}) | ε_{i} (days^{−}^{1}) |
---|---|---|---|---|---|

1 | 0.10669 | 0.01824 | 5.85 | 0.2 | 0.1249 |

2 | 0.01164 | 0.01609 | 0.72 | 0.0091 | 0.02773 |

3 | 0.01762 | 0.0005077 | 34.7 | 0.0333 | 0.01813 |

4 | 0.00436 | 0.26730 | 0.016 | 0 | 0.27166 |

Temperature | σ_{a} (days^{−}^{1}) | μ_{a} (days^{−}^{1}) | μ_{f} (days^{−}^{1}) | f (eggs × days^{−}^{1}) |
---|---|---|---|---|

16˚C | 0.02615 | 0.01397 | 0.03642 | 0.69714 |

28˚C | 0.11612 | 0.06001 | 0.02877 | 8.29500 |

and

The contribution of the first stage of quiescence eggs in all cases, as shown in

Summarizing, assuming that

Up to now the model was discussed considering constant values for the model parameters. But, the effect of

Critical | _{2}) | _{2}, ε_{3}) | |
---|---|---|---|

^{−}^{1}) | 0.0144 | 0.00052 | dnc |

^{−}^{1}) | −0.102 | −0.0037 | dnc |

^{−}^{1}) | −0.01 | 0.00144 | 0:0396 |

^{−}^{1}) | 0.0175 | dnc | dnc |

^{−}^{1}) | −0.00247 | dnc | dnc |

^{−}^{1}) | −0.0597 | 0.420 | dnc |

Values | Stage 1 | Stage 2 | Stage 3 | Stage 4 | Q_{0} |
---|---|---|---|---|---|

31.21 | 18.49 | 4.946 | 0.1502 | 54.787 | |

_{2}) | 31.21 | 2.433 | 18.05 | 0.5481 | 52.230 |

_{2}, ε_{3}) | 31.21 | 2.433 | 48.02 | 0.0529 | 81.710 |

quiescence eggs in the size of mosquito population becomes important in tropical (wet and dry seasons) and temperate (rigorous winter seasons) regions.

The experiments carried out by Silva and Silva [

In a favorable abiotic conditions (high

A mathematical model encompassing four quiescence stages was analyzed. From the model, it was concluded that under certain conditions, the quiescence eggs can improve the fitness of A. aegypti population.

The capacity of the A. aegypti eggs being stored during hostile abiotic factors and, then, hatching to larvae in favorable season with increased fitness, is essential to sustain A. aegypti population to face seasonality. When the quiescence eggs having approximately 120 days are allowed to hatch, these eggs presented the most producible capacity to originate larvae [

Another aspect is regarded to dengue transmission, due to the possibility of the eggs infected with dengue virus sustaining dengue epidemics [

Quantitative analyses, such as the dynamical trajectories, are left to further work. For instance, the dynamical trajectories of the system of Equations (1) are obtained numerically considering the initial conditions, at t = 0, given by

Seasons | Stage 1 | Stage 2 | Stage 3 | Stage 4 | Q_{0} |
---|---|---|---|---|---|

Summer (high) | 78.04 | 1.531 | 6.4 × 10^{−}^{3} | 7.1 × 10^{−}^{7} | 79.57 |

Summer (low) | 0.404 | 3.933 | 2.412 | 1.328 | 8.076 |

Winter (high) | 5.122 | 0.101 | 4.2 × 10^{−}^{4} | 4.6 × 10^{−}^{8} | 5.223 |

Winter (low) | 0.027 | 0.258 | 0.153 | 0.087 | 0.530 |

which corresponds to the introduction of one mosquito in a previously uninfested region. In other words, the equilibrium point before the introduction of infectious case is given by

The author(s) declare(s) that there is no conflict of interests regarding the publication of this article.

Projeto temático FAPESP grant 09/15098-0.

The model described by system of Equations (1) can be simplified joining the quiescence and hatchable stages. For instance, calling

is approximated as

for

where the prime was dropped out.

Defining the average period of time that eggs stay at stage i

with

for

The first equilibrium is the absence of mosquito population, the trivial equilibrium

The non-trivial equilibrium

where the coordinates written in terms of the previously defined parameters are

and the number of adult mosquitoes

The basic offspring number

with

which is the overall production of larvae by all compartments of eggs. Clearly, the non-trivial equilibrium point is biologically feasible if

Letting

where

and

where

with

Notice that

Conversely to the general model, the simplified version of the model does not allow the change in the critical values