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It is set and analyzed a simulation model based on non-linear ordinary differential equations to interpret the dynamics of the microcephaly incidence caused by the Zika virus in a risk group of pregnant women. This one is induced by a population of men in a reproductive age. Also, some parameters of the model were estimated for an average temperature of 23℃ and by using the adjusted functions from references [1] [2]. All system simulations were done with the Maple software and parameters values obtained from several sources (estimated, hypothetic, and form literature). It has been found that the application of contraceptive measures impacts the population of sexually active women. As result, the exposure to congenital abnormalities increases, particularly, microcephaly.

The Zika virus (ZIKV) is transmitted by mosquitoes of the Flaviviridae family of the Flavivirus gender. These ones are pretty close to the ones that transmit encephalitis, yellow fever, dengue, and virus del Nilo. It was isolated in 1947 from blood of the Rhesus monkey (Macacamulatta) that it was feverish during a study of the yellow fever at the Zika forest in Uganda [

ZIKV infection variates in humans, the most common symptoms are: flu, headache, anorexia, eye ache, general discomfort, myalgia, arthralgia, asthenia, edema and diarrhea, that is to say, the symptomatology is unspecific. All this clinical picture may be confused with the other existent diseases, particularly with the most frequent ones, Dengue and Chikungunya [

The ZIKV introduces questions about the possible role of the infection with the microcephaly; actually there are only ecological evidences. Microcephaly is defined as the presence of a cranial perimeter less than a couple of standard deviations for the average age, sex, and gestational age [

In the last years it has been observed an increase in the infectious diseases, this is due to the known symptoms as well as the reemerging ones that have appeared with a high incidence because of social, economic and health factors. Such is the case of ZIKV, an endemic virus in some countries of Africa and Asia that has been expanded to other zones [

Recently, some reports treating this issue have appeared. Daozhou et al. (2016), have analyzed a mathematical model for the prevention and control of Zika, including the vectorial and sexual transmission, with data of epidemic outbreaks of Brazil, Colombia and the Salvador. From those studies they conclude that the R_{0} is more sensitive to the bite rate and the natural death rate of the mosquitoes as well as the sexual transmission increases the risk of infection, the epidemic size and extends the outbreak [_{0} of the ZIKV infection, as an indicator of potential transmission, using data from the South Pacific region [

The model proposed in this paper helps to provide information that may help to elaborate effective control strategies, potential scenarios and transmission mechanisms of the virus.

It is proposed a simulation model based in non-linear ordinary differential equations to describe the dynamics of pregnant women that are susceptible to get infected by ZIKV and may present fetal microcephaly induced by this virus. The variables of the model are described in

Variable | Description | Initial populations |
---|---|---|

The susceptible and sexually active women’s population | ||

The susceptible pregnant women’s population | ||

ZIKV positive pregnant women’s population | ||

Infected pregnant women with no development of microcephaly | ||

Infected women’s population | ||

Susceptible and sexually active men’s population | ||

ZIKV positive men’s population | ||

The non-carrier Aedes aegypti mosquitoes population | ||

The virus-carrier Aedes aegypti mosquitoes population | ||

Total population (men + women) | ||

Total population of sexually active women | ||

Total population of sexually active men | ||

Total population of mosquitoes |

Parameter | Description | Estimated/hypothetic value |
---|---|---|

The fraction of women that get into the sexually active population | 30 | |

Men’s fraction that get into the sexually active population | 20 | |

The natural mortality rate of persons | 0.0003 | |

Infected pregnant women’s fraction that are not developing microcephaly | 0.1 | |

Infected pregnant women’s fraction that develops microcephaly | 0.9 | |

Developing rate of the infection in the pregnant women | 0.7 | |

Transmission probability to the susceptible women | 0.7913 | |

Probability of pregnancy | 0.6 | |

The infected women recovery rate | 0.01 | |

Fraction of women that use contraceptive measures | 0.3, 0.6, 0.8 | |

Transmission probability to the susceptible men | 0.7913 | |

Transmission probability to the non-carrier mosquitoes | 0.6, 0.35 | |

Death rate of the mosquitoes | 0.035 |

In the flux diagrams of the Figures 1-3,

The dynamic system in the sexually active women that get pregnant, susceptible to ZIKV as well as exposed to microcephaly are (according to

In this system

From these equations (6) and (7)

In a similar way, Equations (8) and (9) are result of inflows and outflows.

The variation in time of each total population

As well,

Then,

Varying with respect to t, it is obtained

In the case of mosquitoes,

Varying in function of t,

From the dynamic systems, replacing the equations (1)-(7) in (10)-(13), it is obtained the dynamic system of the populations,

The dynamic systems (1)-(5), (6)-(9) were simulated using the Maple software, with the values of

In

fraction under contraceptive protection). The behavior of the susceptible pregnant women population rise showing a maximum point that is each time smaller as h increases, namely, with the contraceptive measures, the susceptible women population decreases. A similar behavior is presented by the infected pregnant women. Both populations tend to stabilize in an average of 35 days.

The pregnant women’s population that develops microcephaly have a quick growth till almost 25 days, after that, the population tends to have a linear-like growth through time as h increases. However, this population has a maximum point when

The behavior of the simulations shows that the susceptible pregnant women’s population is sensitive to the women’s fraction under birth control. This is an expected result if we account for an 80% of this fraction, were the susceptible women, infected pregnant women and women without microcephaly development population’s decrease. With this model it is possible to make predictions and analysis to design more effective strategies in the future.

ZIKV has a little known natural history, then, it is primordial to generate new mathematical studies to gain a better understanding of this disease and propose effective control strategies.

The proposed model in this paper can be applied (with the respective adjustments) in the study of other viral infections transmitted by vectors that generate risk of other pathological complications.

AML thanks to Grupo de Modelación Matemática en Epidemiología (GMME), Facu- ltad de Educación\’on, Universidad del Quindío-Colombia.

Pizza, D.M.M., Loaiza, A.M., Arias, O.A.M., Sossa, V.A., Muñoz, C.A.A., Osorio, S.R., Contreras, H.M., Montoya, J.F.A., Patiño, G.O., Contreras, I.P., Perea, M.E.C., Guerra, M.A.E. and García, J.A.V. (2016) A Simulation Model for the Risk of Fetal Exposure Originated by the Zika Virus (VIZK). Health, 8, 1178-1186. http://dx.doi.org/10.4236/health.2016.812121