On February 1^st, 2016, the World Health Organization declared ZIKV epidemic in the Americas as a Public Health Emergency of International Concern. The foremost reason is due to an emerging linkage with congenital birth anomalies such as microcephaly and Guillain-Barre syndrome [2] plus mode of transmission which had not known before (sexual-transmission). Since then, ZIKV is one of the most uprising viral disease in the world, especially in the Americas. Resent studies report that ZIKV is transmitted not only via mosquito bite but also via sexual contacts, blood contamination and mother-to-child-transmission [1] . The mosquito involved in the transmission is infected Aedes Aegypti. This particular mosquito belongs to the family Flaviviridae and Genus Flavivirus. Chikungunya, Dengue and Yellow fever are also the off shots of this family. Prenatal or perinatal complications of ZIKV infections have been noted due to ZIKV. There is some evidence that perinatal transmission may occur, most probably transplacental or during the delivery of a viraemic mother [3] . The virus has been detected in human saliva, semen and urine of which leads to sexual-transmission of ZIKV [4] [5] .

Despite the fact that attack of ZIKV shows moderate disease symptoms, like that of the clinical presentation and transmission cycles in the epidemiology of Chikunganya and Dengue, symptoms of ZIKV disease encompasses conjunctivitis, skin rash, fever, muscle pains in various joints, non comforting nuisance and headache, which can last normally between 2 to 7 days time period. Not limited to those, it has been found that about 80% of individuals infected by ZIKV are asymptomatic [6] [8] . In new babies, congenital microcephaly is a descriptive diagnosis. According to WHO, it is defined as a head circumference equal to or lower than two standard deviations below the mean () for age and sex or about less than the second percentile [9] . Microcephaly results from reduced brain growth. Affected infants are born with a smaller head size than normal, or their head stops growing after birth. It is a rare condition with estimated incidences varying geographically; in the United States reported incidence ranges from 2 to 12 babies per 10,000 live births. There is an association between the severity of microcephaly and that of neurological impairment [7] [10] .

The causes are numerous, including chromosomal abnormalities, intrauterine infections, exposure to toxic substances during pregnancy, and severe malnutrition [11] . Some countries and territories in the Americas with reported congenital syndrome associated with Zika virus infection as of July 14, 2016 (Figure 1). WHO estimates that the fetuses of one in 100 women with Zika will go on to develop microcephaly or other abnormalities [12] .

No specific antiviral medication is available for ZIKV disease. Treatment is generally supportive and can include rest, fluids, and use of analgesics and antipyretics. Aspirin including other nonsteroidal anti-inflammatory drugs should be avoided until dengue can be ruled out to reduce the risk of hemorrhage. Febrile pregnant women should be treated with acetaminophen. Persons infected with ZIKV, dengue, or Chikungunya virus should be protected from further mosquito exposure during the first few days of illness to reduce the risk for local transmission [6] .

The very first incidence of ZIKV was identified in 1947 from species of monkey in the African forest called Zika from Uganda. And the first isolated case identified from humans in 1952 also from Uganda and then later from a neighbor country, the United Republic of Tanzania. However, comprehensive genomic comparison showed different sub-classes reflecting the existence of two main lineages, one African and the other from Asia lineage [13] [15] . ZIKV has been shown intermittent disease patterns among the people of Africa and Asia. Little clinical importance was attributed to ZIKV, for the reason that only few symptomatic cases were reported in some African and Southeast Asiatic countries at the time [14] . This situation changed in 2007, when a large outbreak was registered on the Yap Island, Micronesia, caused by the Asian ZIKV lineage. Moreover, the outbreaks reported beginning this era are from the Pacific in 2007 and 2013 from French Polynesia and Yap Island (i.e. Federated States of Micronesia) [16] . There were subsequent incidents of the virus spread to other areas of the region like Cook Islands, Chile, Pacific Islands including New Caledonia, Fiji, Samoa, Solomon Islands and Vanuatu (Figure 2).

ZIKV has been steadily increasing its geographical domain, ever since its initial appearance. The pace of ZIKV outbreaks as epidemic in the Americas is on rise. Similarly, the spread of this monster is reported in more regions of the world that were previously completely unaffected, that includes Europe [18] . The immunity of Europeans population is relatively immature as contrast to Afro-Asian Populations. At the arrival of summer season in the northern hemisphere is the most suitable time when Aedes mosquitoes in Europe find optimum climate to get infected by ZIKV and later transmit this virus to tourists and then into local population through biting [17] . At this point, it is relevant to cover and analyze the monthly traffic of travelers by air, arriving into the cities of Europe from the Zika affected regions of both the American continents and Europe.

Brazil’s first confirmed Zika infection was in March 2015. Over five years preceding 2015, the country saw between 130 and 170 cases of microcephaly each year [20] . In the first nine months of 2015, this figure roughly doubled. By the end of 2015, over 2400 further similar cases had been reported [19] . Up to February 27, 2016 according to the Brazilian Ministry of Health, 5640 cases of microcephaly in the country have been reported, and 583 of them are confirmed to have microcephaly and/or other central nervous system findings, suggesting of congenital infection. Out of the total cases with confirmed microcephaly, only 67 fulfill the laboratory criteria for ZIKV infection, and 4107 across Brazil remain under investigation [14] .

Regarding general ZIKV infections, data announced on December 2015, by the ECDC estimate that between 440,000 and 1.3 million cases by autochthonous transmission of ZIKV have occurred in Brazil (Figure 3), but these data are largely underestimated and the real magnitude of the Brazilian ZIKV epidemic may be much higher [14] . As of June 2016, around 500,000 Zika virus disease (ZVD) cases have been estimated in Brazil, and autochthonous circulation has been observed in 40 countries from the Americas [8] .

Basing on the nature of epidemic, though with a relatively long history since its knowledge in 1947, few researches have been conducted. Probably, the main reason is that its effects where not vividly clear to human being as of now. However, this does not rule out that in the past people had not been affected by the problem. About three years ago, a number of researches have been working to explore the disease down so as to come up with some means to control it. According to Rojas DP et al. [8] all age groups are vulnerable, but the most affected age group is 20 to 49 years of age. A similar study previously published outbreaks in Yap Island, Micronesia, and in Salvador, Brazil [21] . As the population was fully susceptible to ZIKV transmission before the outbreaks, it is expected that all age groups would be affected. Joacim Rocklov et al. [2] , studied which seasons in the year are more risk for ZIKV transmission. There are as well

more researches in medicine and other discipline. This paper is more concerned with mathematical models exploring the epidemic. One of the model was done by Adam J. Kucharski et al. in early 2016 [22] . The model provided understanding ZVD in mathematical perspective, basing on some assumptions the model did not include sexual transmission. Khalid et al. in the late of 2016, have formulated the mathematical model by including sexual transmission [23] . One side this article is an improvement of Kucharski’s model modified to include sexual-transmission. However, not only the later but also infections to newborns through MTCT. Therefore, the model is a simple SIR model that includes newborn as part of the infected class.

For endemic equilibrium, ,. The system of Equation (3) reduces to

Solving this simultaneously, the equilibrium point is: (5)is obtained from the equation: (6)The equilibrium point reduces to a disease free when.4.1.1. Existence of the Endemic EquilibriumConsider Equation (6):. Assign. Endemic equilibrium exists whenever there is a positive solution (i.e. value of). When,.When,. Therefore at some point in the interval there exists, which proves the existence of endemic equilibrium. As an example, consider a simulation: Choose any; say

. Then. That is, the value for is (Figure 5).

The graphs in Figure 6 depict the dynamics of ZIKV for the first 10 years in Brazil starting June 2016. The graphs for I and R classes are not clearly seen in Figure 6(a), therefore can be seen in Figure 6(b). Although initial number individuals in a recovery class may be lower than infectious class (Figure 6(b)), but the prevalence of recovery class grows faster than the infected class. In the later time of the epidemic there more individuals in the recovery than infectious class.

For M estimations, the differential equation from the model (8) is

simulated for 25 months (Figure 7). In this case the objective is to estimate the

total number of children who has been inflicted by ZIKV. It is clear that this accounts for all children including the initial amount as of January 2016. Up to February 27, 2016 according to Brazilian Ministry of Health, out of all reported microcephaly cases only 67 fulfill the laboratory criteria for ZIKV infection [14] .

Then, the February 2016 cases of microcephaly is assumed to be initial value for January 2016 in this simulation. Estimation for all newborns regardless of those who die later after birth, so long as had been diagnosed with microcephaly linking to ZIKV then are part of this integral. After 25 months since January 2016, the number of ZIKV infected newborns in the same month 2017 is approximated above 2100 (Figure 7(a)) and expected to rise up about 4350 in January 2018. The plot of M is an accumulation curve (it is not prevalence), (Figure 7(a) and Figure 7(b)). The graph of M is not a straight line, but is an increasing function (Figure 7(b)).

as in Figure 8(a), infectious class growth is accompanied with a silent recovery class escalation. Note that there is a great number of individuals 80% who are asymptotic to the disease [6] [8] , of which they may recover without their knowledge this is not accounted for in this case. What is considered are recovery individuals who where once diagnosed with ZIKV infection. Also the prevalence for recovery is called silent simply because generally no one cares when an individual is recovered, therefore there prevalence is unnoticed. Later the recovery class R may increasingly out number the size of population in the Infectious class I (Figure 6(b)). Whenever increases which leads to increase in the infections, number of individual recovering also increases considerably maintaining the recovered population size more than the infectious class (Figure 8(b)).

At the latest, if the situation is not attended or interrupted, naturally may lead to

equilibrium as in Figure 9. The recovered class has two important roles in suppressing the epidemic. One is that the recovered individuals build up immune against ZIKV infection. The other reason is that since the higher density of recovered individuals is in the most affected regions, this minimizes interactions between susceptible and infectious, therefore reducing spread of the epidemic. So when more individuals are recovered compared to infectious (Figure 9(a)), natural resistance is strengthened. This may save to take the endemic equilibrium point of the epidemic at a state where there is very few infectious of ZIKV for long time. This may explain a situation in the countries with historic start of ZIKV like Africa and Asia, particularly in Uganda and Tanzania. In such regions occasionally, new born with microcephaly may happen and is likely to be ignored so long as there is no substantial outbreak of the epidemic. Basing on the model simulation such a situation is reach around one generation time since start of the epidemic.

Among other reasons that can lead to microcephaly situation side track the societies attention to ZIKV disease outbreak especially when the infections are low and unnoticeable is not keeping records, especially in most villages of the developing countries. Therefore, appreciation should be given to the Americas, like Brazil for keeping records of microcephaly born children which is one of the reasons lead to notice the break of epidemic [20] .

According to the model (2), two main factors to be controlled in order to diminish or eliminate the epidemic basing on the model include reducing both unsafe sexual-interaction and mosquitoes bites against humans. Controlling the two factors to the extent that such as may avert the growing epidemic to one dying (Figure 10(a)).

The model (2) estimates that in 15 years for the region can be free of the epidemic, thus leaving growth of susceptible class growing at its natural pace for Brazil Figure 10(b). It has been found that contribution in the spread of ZIKV infections by sexual-transmission is in low percentage compared to human-mosquito interaction [25] [26] . Therefore, effective elimination of the epidemics is basically freeing individuals from mosquitoes bites. This include getting rid off still water bodies, use of mosquito nets, insect repellents and use of insecticides to help exterminate mosquitoes cultures. Nevertheless, sexual-transmission has to be controlled as the second adverse case of transmission. That is, avoid or have safe sexual-interactions, especially with people traveling to and from infected regions. This may involve use of condoms, circumcision and all that contribute to secured sexual-interaction [26] .

As for women who want to get pregnant but are not sure of ZIKV infections (no symptoms) but they have been exposed to ZIKV infected regions or had sex with ZIKV diagnosed partners, should wait at least eight weeks after exposure. For those with diagnosed ZIKV infections, wait after eight weeks since the start of symptoms (Figure 11).