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Nowadays the Zika virus (ZIKV) has been one of the most studied vector-borne diseases due to the considerable outbreaks that have generated around the world as well as due to the new transmission mechanisms and health complications originated. According to statistics of the INS-Colombia for July 2016, 68% of the population infected by ZIKV (confirmed cases) are pregnant women. Furthermore, the Quindío department belongs to the states with more than 50% of the total infected persons being pregnant women. Taking into account those characteristics, a theoretical model is proposed and analyzed to describe the population dynamics considering the sexual and vectorial transmission of ZIKV, with special emphasis in the consequences of the non-vectorial transmission in the population. The obtained results with simulations through the beta parameter indicate that the probability of sexual transmission between susceptible women and infected men points out the importance of campaigns to inculcate prevention measures for the safe sexual relationships between ZIKV infected population.

Recently the concern caused by the Zika virus (ZIKV) has increased, and this infection has been turned in a public health problem in several countries of the American continent. This is mainly due to the relationship with diseases such as: microcephaly and Guillain Barre Syndrome [

It is well known that ZIKV is a disease transmitted by the bite of an infected mosquito (principally the Aedes aegypti gender). There are secondary ways of transmission, although the knowledge of them is limited. The virus has been detected in human saliva, semen and urine [

Particularly in Colombia, according to the statistics in the “Boletín Epidemiológico Semanal del Instituto Nacional de Salud” [

country with the highest quantity of infected pregnant women (confirmed cases). We divide those percentages into three main groups, each group (sixteen states) with more than fifty percent of infected expectant women (58% - 87%). Then, it is especially important to study and analyze the sexual transmission in those states. Since the Quindío department is one of the most affected by this issue, we will consider it in our investigation.

Taking into account these facts, we propose a model in R^{3} which considers the vectorial transmission as well as the sexual transmission of the ZIKV. The basic reproduction number and the stability analysis of the system are obtained. After that, the simulations using the Maple software are carried out to observe the behavior of the epidemic threshold as function of several parameters, the behavior of the sensitivity indexes as well as the transmission probability by sexual way in the human populations. Unlike the model proposed by Gao et al. [

It is important to model this dynamics to offer useful information to the public health care institutions in the country, for the application of preventive measures to the sexual transmission of this virus, this is due to the potential propagation of this disease and the collateral effects that may cause. Note that in Colombia almost all the states present high percentages pregnant women infected by ZIKV (see

It is proposed a simulation model, in R^{3}, based in non-linear ordinary differential equations following the Sir Ronald Ross formalism to describe the dynamics of the sexual and vectorial ZIKV transmission, accounting the following assumptions: closed populations, variations in time of the ZIKV infected pregnant women population, infected men populations, and the virus-carrier A. aegypti mosquitoes, and the constant recovery rate of infected women and infected men. Also, it is considered the ZIKV transmission from the infected men to the susceptible women, assuming promiscuity.

The variables of the model are:

The parameters of the model are shown in

The differential equations system to describe the infectious process is (

Parameter | Description |
---|---|

Transmission probability by sexual contact between susceptible women and men infected by ZIKV | |

Transmission probability by the bite of virus-carrier mosquitoes to susceptible women | |

Recovery rate of infected people | |

Natural mortality rate of the human population | |

Transmission probability by the bite of virus-carrier mosquitoes to susceptible men | |

Transmission probability to non-carrier mosquitoes by the bite to the infected women | |

Transmission probability to non-carrier mosquitoes by the bite to the infected men | |

Mortality rate of the mosquitoes |

where

And, the region where the system trajectories have epidemiological sense is,

The stability analysis begins calculating the equilibrium points by solving the algebraic

system, which is associated to the non-linear differential equations,

Obtaining

Substituting (7) in (4), it is found:

Simplifying the previous equation, we have,

Replacing (7) and (8) in (6), we obtain:

where

For

Simplifying Equation (10), it is derived the following equation,

where,

Due to A > 0 and if we assume that at least one of the B, C and D is less than zero, it guarantees a change of sign in the coefficients of the quadratic equation. Following the signs rule of Descartes it is obtained a positive and real root (

In the linearization process for the non-lineal equations system (1)-(3), is calculated the Jacobian matrix at the generic equilibrium point

Evaluating the Jacobian matrix in the free of infection equilibrium point, we obtain:

So, the characteristic equation

where,

Under the Routh-Hurwitz criterion, it should be accomplished that

At the equilibrium point

where,

Solving by the first column and cofactors, we have,

with

Applying the Routh-Hurwitz criterion, this equation has 3 roots with negative real part if the inequalities are met

The so called epidemic threshold

Then doing

so,

where;

For a best understanding of the basic reproduction number

The terms

The term

For an average annual temperature in Armenia-Quindío, Colombia of 19.5˚C, were estimated the transmission probabilities

where

The local stability analysis is carried out using the estimated parameters. We start with the calculation of the free of infection and prevalence equilibrium points, which are determined doing the differentiation of the system (1)-(3) equal to zero, and solving the non-linear algebraic system for each demographic variable.

Using the MAPLE software the equilibrium points are determined.

・ For

・ For

・ For

With information in the

Another threshold is obtained using the local sensitivity analysis, this one represents a relative measure of the change in one variable when the value of one parameter changes

Parameter | ||||||||
---|---|---|---|---|---|---|---|---|

Value | 0.5178 | 0.14 | 0.0003 | 0.5178 | 0.52872 | 0.52872 | 0.03604 | 0.3 - 0.6 - 0.8 |

Equilibrium point | Eigenvalues | Stability | |
---|---|---|---|

0.3 | (0, 0, 0) | 0.7157, −0.3032, −0.7290 | Unstable |

0.3 | (0.838, 0.779, 0.959) | −0.6254, −0.8988, 1.2636 | Stable |

0.6 | (0, 0, 0) | 0.7669, −0.5418, −0.5418 | Unstable |

0.6 | (0.873, 0.779, 0.960) | −0.6273, −0.9166, −1.2634 | Stable |

0.8 | (0, 0, 0) | 0.7971, −0.5569, −0.5569 | Stable |

0.8 | (0.888, 0.780, 0.960) | −0.6278, −0.9249, −1.2639 | Stable |

[

where,

The results of the sensitivity analysis of

In

Parameter | ||||||||
---|---|---|---|---|---|---|---|---|

0.758 | 0.241 | 0.241 | 0.758 | −1 | −1.513 | −0.003 | 0.5167 |

to the most sensitive parameters.

In

The simulations of system (1)-(3) were obtained using the values of the parameters as reported in

From the results of the simulations, which are depicted in

It is possible to see that upon beta increases, the infected women population increases in a time t. Then, it is extremely important to apply prevention measures in the sexual relationships of the infected population, and then help to stop the virus propagation at large scale. Together with these measures, it is expected to have a decrease in the diseases derived by this virus, as congenital microcephaly and Guillian-Barre syndrome.

The basic reproduction number is inversely proportional to the recovery rate of the infected persons and the mortality rate of the mosquitoes. That is to say, it is not appropriate to decrease the values of these parameters by treatment and control of the mosquito.

Actually, the Ross-Macdonald formalism is very important to model Vector-Host diseases as Malaria, Dengue, Chagas, Chikungunya and ZIKV, etc. In future works it is important to carry out the goblal stability analysis of the model and add the horizontal transmission in the man as well as the model without considering Sir. Ronald Ross formalism.

AML thanks to Grupo de Modelación Matemática en Epidemiología (GMME), Facultad de Educación, Vicerrectoría de investigaciones, Universidad del Quindío-Colombia.

Manrique, O.A., Pizza, D.M.M., Loaiza, A.M., García, J.A.O., Muñoz, C.A.A., Osorio, S.R., Osorio, A.J., Contreras, H.M. and Montoya, J.F.A. (2017) A Simulation Model for Sexual and Vec- torial Transmission of Zika Virus (ZIKV). Open Journal of Modelling and Simulation, 5, 70-82. http://dx.doi.org/10.4236/ojmsi.2017.51006