Prevention and Control of Zika as a Mosquito-Borne and Sexually Transmitted Disease: A Mathematical Modeling Analysis

The ongoing Zika virus (ZIKV) epidemic in the Americas poses a major global public health emergency. While ZIKV is transmitted from human to human by bites of Aedes mosquitoes, recent evidence indicates that ZIKV can also be transmitted via sexual contact with cases of sexually transmitted ZIKV reported in Argentina, Canada, Chile, France, Italy, New Zealand, Peru, Portugal, and the USA. Yet, the role of sexual transmission on the spread and control of ZIKV infection is not well-understood. We introduce a mathematical model to investigate the impact of mosquito-borne and sexual transmission on the spread and control of ZIKV and calibrate the model to ZIKV epidemic data from Brazil, Colombia, and El Salvador. Parameter estimates yielded a basic reproduction number 0 = 2.055 (95% CI: 0.523–6.300), in which the percentage contribution of sexual transmission is 3.044% (95% CI: 0.123–45.73). Our sensitivity analyses indicate that 0 is most sensitive to the biting rate and mortality rate of mosquitoes while sexual transmission increases the risk of infection and epidemic size and prolongs the outbreak. Prevention and control efforts against ZIKV should target both the mosquito-borne and sexual transmission routes.

cases were reported in El Salvador 9 . The WHO declared the epidemic a Public Health Emergency of International Concern (PHEIC) on February 1, 2016 10 , and the U.S. CDC's Emergency Operations Center has moved to the highest level of activation on February 3, 2016 11 . Based on the reported dengue data from 2015, WHO estimated that up to four million people in the Americas could be infected by ZIKV in 2016. Without effective intervention, the situation has considerable potential to worsen, due in part to the upcoming 2016 Summer Olympics in Rio de Janeiro as well as anticipated mosquito abundance increases caused by an ongoing El Niño.
ZIKV has been detected in serum, saliva, urine, and semen [12][13][14] . It has also been detected in urine and semen even after it disappears from blood 15 , and in one convalescent case it was detected in semen 27 and 62 days after onset of febrile illness 14 . Indeed, recent studies show that ZIKV can be transmitted via sexual contact. In 2011, it was reported that an infected male had infected a female by having vaginal sexual intercourse, even before his onset of symptoms 16 . After the confirmation of the first case of sexually transmitted ZIKV of the current outbreak in Dallas County by the CDC on February 2, 2016 17 , six more confirmed and probable cases of sexual transmission of ZIKV in the U.S. were reported by CDC on February 26, 2016 18 , and Europe's first case of sexually transmitted ZIKV was diagnosed in France in Febraury 2016 19 . A case of ZIKV infection imported in Florence, Italy ex-Thailand, leading to a secondary autochthonous case, probably through sexual transmission in May 2014 was retrospectively diagnosed in 2016 20 . Since 2015, Zika infections likely acquired through sex have been reported in Argentina, Canada, Chile, France, Italy, New Zealand, Peru, Portugal, and the USA 21 .
The study of the ZIKV outbreak on Yap Island 1 indicates that cases occurred among all age groups, but the incidence of ZIKV disease was highest among persons 55 to 59 years of age with the mean age of 36 years and 61% female. Since ZIKV infections are mostly asymptomatic or have mild symptoms lasting two to seven days, the disease has little impact on sexual activity 18 . If ZIKV is sexually transmissible, then it is necessary to abstain from sexual activity or consistently use condoms during convalescence. CDC has recently issued interim guidance on safe sex during a Zika outbreak 22 . This is particularly important to pregnant women in areas where the ZIKV is circulating.

Results
Modeling. Mathematical modeling has become a crucial tool in designing prevention and control measures for infectious diseases 23,24 . A recent study 25 used a mosquito-borne model to examine the 2013-14 outbreak of Zika on the six major archipelagos of French Polynesia. To investigate the role of sexual transmission in the spread and control of Zika virus disease, we developed a deterministic model of Zika disease transmission that takes into account both mosquito-borne and sexual transmission modes (Fig. 1). Symptomatically infected humans are contagious to both mosquitoes and humans during the incubation period that is typically between 2 and 7 days. This is because the viremia and virusemenia occur before the end of the incubation period, although the viral load of exposed (presymptomatic) people may be lower 16 . After this period, infected humans develop symptoms. Symptomatic humans are more contagious to mosquitoes than exposed individuals and are also able to transmit the virus to partners through sex 18 . The virus appears to persist longer in semen and urine than in serum 14,15 . Following the period of viremia, symptomatic humans enter the convalescent stage and can no longer infect mosquitoes. However, such individuals remain infectious to humans, though with reduced infectivity. The infected humans' convalescent period ends with lifelong immunity. Sexual transmission of ZKIV from asymptomatically infected humans has not been documented, so they are assumed to be noninfectious to humans. The timescale of human demography is far longer than that of the epidemiological dynamics, so we ignore human births and deaths when modeling an outbreak.
We make the following additional assumptions: (i) Mosquitoes cannot be infected by biting asymptomatically ZIKV infected people; (ii) The sexual ratio of humans is 1:1 and male and female are subject to almost the same epidemiological factors; (iii) The end of the viremic period coincides with the disappearance of symptoms in symptomatically infected individuals (see Fig. 1). An individual may progress from susceptible (S h ) to asymptomatically infected (A h ) to recovered (R h ), or exposed (E h ) to symptomatically infected (I h1 ) to convalescent (I h2 ) to recovered (R h ), while a mosquito may progress from susceptible (S v ) to exposed (E v ) to infectious (I v ).
Parameter estimates. All parameter descriptions and ranges are summarized in Table 1. To parameterize our model, we used reasonable epidemiological parameters based on our current understanding of Zika epidemiology and transmission dynamics. Zika virus and dengue virus are arboviruses of the same genus Flavivirus spread by mosquitoes of the same genus Aedes and have similar symptoms, high proportion of asymptomatic infections, duration of incubation and infectiousness 7 . Hence they have the same number of bites on humans per mosquito per unit time, a, and we anticipated that their transmission probabilities per bite from mosquitoes to humans, b, and from humans to mosquitoes, c, respectively, are comparable.
Many parameters of ZIKV infection are not available, but we can make some reasonable assumptions to estimate specific measurements of the infectivity. For instance, consideration of other human sexually transmitted infections may provide an initial basis for analysis. We note that gonorrhea appears to display a high probablity of transmission per coital act (results ranging from 0.19-0.65, with male to female transmission higher than female to male 26 ). Others show a considerably lower transmission probability per coital act, such as HSV-2 (0.0005) 27 .
The average sexual frequency over sexually active ages is twice a week and the frequency of sexual intercourse over all age groups is assumed to be once a week 28 . The sexual transmission rate of symptomatically infected people (transmission probability × contact rate), β, is assumed to range from 0.001 to 0.10, which means the transmission probability per sex act is between 0.007 (mild infectivity) and 0.70 (severe infectivity).
Fitting Zika data in Brazil, Colombia and El Salvador. To use our model to fit the reported ZIKV cases in Brazil, Colombia, and El Salvador (see Fig. 2(A)), we assumed that the three countries share common parameter values (see Table 1), except for country population size and initial conditions (see Table 2). Since large scale mosquito-control campaign has been taken in these Zika affected countries, we assumed that the ratio of mosquitoes to humans m is time-dependent and used a cubic spline function of time with n m parameters to describe m(t). Figure 2(B) demonstrates that our model provides good fits to the reported Zika data from Brazil, Colombia, and El Salvador up to February 27, 2016 which is represented by the right vertical axis. In Brazil, the outbreak started in the spring of 2015, has passed its peak, and seems under control for the time being. In Colombia and El Salvador, the disease started in the summer of 2015 and is reaching its peak now. More Zika, GBS and microcephaly cases are expected from other countries in South and Central Americas and the Caribbean. The starting time and geographic spread of Zika ( Fig. 2(A)) indicates that it is following the path of dengue and chikungunya and has the potential to be introduced to many other countries where the Aedes species mosquitoes are competent, including some southern states in the U.S. Table 1, we used the Latin hypercube sampling method 29 to generate 5,000 samples by assuming a uniform distribution for each parameter, and calculated the corresponding uncertainty on the basic reproduction numbers of either mosquito-borne transmission or sexual transmission or both. The median and confidence interval of the distribution of the basic Relative human-to-mosquito transmission probability of exposed humans to symptomatically infected humans. (Dimensionless)

Estimation of the basic reproduction number. Based on parameter ranges in
Transmission rate from symptomatically infected humans to susceptible humans. (Per day) 0.001-0.10 0.05 Assumed κ: Relative human-to-human transmissibility of exposed humans to symptomatic humans. reproduction numbers (see Fig. 3(A)) are 2.055 (95% CI: 0.523-6.300) for 0  , 1.960 (95% CI: 0.450-6.227) for hv  , and 0.136 (95% CI: 0.009-0.521) for  hh , respectively; the median and confidence interval of the distribution of the percentage of contribution by sexual transmission in  0 is 3.044 (95% CI: 0.123-45.73). This suggests that sexual transmission alone is unlikely to initiate or sustain an outbreak. However, if the human-to-human transmission probability is very high, then its promoting effect on the transmission of ZIKV cannot be neglected.
To identify the key factors that determine the magnitude of the basic reproduction number, we performed global sensitivity analysis (see Fig. 3(B)) with 1,000 random sample uniformly distributed in the range of parameter values from Table 1 and 1,000 bootstrap replicates 30 . The basic reproduction number is most sensitive to mosquito biting rate and mortality rate.
To determine the dependence of the basic reproduction number on the controllable model parameters,   Table 2.
Scientific RepoRts | 6:28070 | DOI: 10.1038/srep28070 Figure 4 indicates that mosquito-control, personal biting protection, and sexual contact protection are all important measures to control Zika virus infection. As season changes, m will change along time which may cause fluctuating outbreaks of Zika in the future. Note that even the vector-control and bitting protection measures are successful, if the human-to-human sexual transmission probablity is very high, then the disease still could persist in the population and the outbreak could be prolonged.
Attack Rate. Attack rate or attack ratio z is the fraction of the population that becomes infected, which is also an important concept in measuring the transmission of infectious diseases.
To identify the key factors that affect the attack rate, we performed global sensitivity analysis (see Fig. 5) with 1,000 random sample uniformly distributed in the range of parameter values from Table 1 and 1,000 bootstrap replicates. The attack rate is also most sensitive to mosquito biting rate and mortality rate.
Numerical Scenarios. In Table 3, selected scenarios from Fig. 4 are presenteded, demonstrating the range of transmission contributed by sexual activity. If m = 5 and a = 0.5, sexual transmission only accounts for 1.351% of total infections at β = 0.01, increases to 6.506% if β = 0.05 and 12.45% if β = 0.10. When β = 0.05, m = 5, a = 0.5, the attack rate is up to 82.65%; it is impossible to contain infection even reducing mosquito population by 55%, though the reproduction number for human-mosquito transmission is less than one in this scenario. In this case, both public and individual disease intervention measures are imperative (e.g., 55% reduction in mosquito  Parameter values are given in Table 1. Scientific RepoRts | 6:28070 | DOI: 10.1038/srep28070 population and 50% reduction in unprotected sex behaviors). In all scenarios The three columns refer to controllable parameters, basic reproduction numbers, and attack rates, respectively.

Discussion
Lacking data on early dynamics to dissect the contributions from different transmission routes of ZIKV, we designed a SEIR-type model based on classic epidemic theory 23 and studied the impact of sexual transmission on the prevention and control of the mosquito-borne ZIKV. As a result, we considered here the exponential growth dynamics of ZIKV infection. In contrast, sub-exponential growth dynamics for sexually-transmission diseases has been reported previously 34 . By using reasonable ranges of parameter values, we obtained some estimates of the basic reproduction number and provided a qualitative analysis of the contribution of mosquito-borne transmission and sexual transmission on the spread of ZIKV.
Sexual transmission increases the risk of infection and epidemic size, but itself may not initiate or sustain an outbreak. Statistically, the transmission contributed by sexual activity is a small percentage of the total transmission in attack rate (4.437 (95% CI: 0.297-23.02)). However, the potential of non-vector-borne transmission could complicate efforts on containing the spread of ZIKV. Under certain circumstances, culling a large fraction of mosquito population such that the basic reproduction number for human-mosquito transmission below one may be not sufficient in the absence of sexual risk reduction.
At the beginning of an outbreak, it is best to contain ZIKV infection by vector control (larviciding and adulticiding) and insect bite precautions (bed net and insect repellents). However, insecticide resistance in Aedes aegypti and Aedes albopictus and poverty in the most vulnerable regions can compromise the success of bite reduction. Subsequently, measures on reduction of sexual transmission shall be implemented. Especially, pregnant women or those at childbearing age who have pregnancy planning need to avoid unprotected sexual contact.
To better understand the transmission mechanisms and estimate the level of sexual transmissibility, solid clinical and epidemiological data are required. However, the cross-reaction of other Flaviviruses and the mild illness of Zika make the infections highly unreported or misdiagnosed. The magnitude of human-to-human transmission may be underestimated if a longer convalescent phase with higher infectivity is confirmed, or overestimated if infected women cannot transmit Zika virus to their sex partners. We assumed a homogeneous mixing human population, although heterogeneities such as gender, culture, religion, and socioeconomics deserve further investigation. Moreover, it is much more difficult to formulate, parametrize, validate and analyze such models.
If sexual transmission were not small compared to vector-borne transmission, there would have been some excess seen in particular age groups (sexually active age 1 ), partnership groups (individuals with regular sexual intercourse), behavioral groups (individuals frequently having unprotected sex), and professional groups (sex workers). A standard questionnaire plus a recent history of sexual contacts for residents in Zika-hit regions will provide these data to examine the mode of sexual transmission.

Methods
Mathematical Model. Human population is divided into six classes: susceptible (S h (t)), exposed (E h (t)), symptomatically infected (I h1 (t)), convalescent (I h2 (t)), asymptomatically infected (A h (t)), and recovered (R h (t)) at time t > 0, and the mosquito population is divided into three classes: susceptible (S v (t)), exposed (E v (t)),  Table 1. and infectious (I v (t)), respectively. N h = S h + E h + I h1 + I h2 + A h + R h denotes the total number of humans and N v = S v + E v + I v denotes the total mosquito population, both are assumed to be constant. We use the SEI type of structure for mosquitoes and SEIR type of structure for humans 23,24 . Based on the assumptions, the ZIKV transmission dynamics between humans and mosquitoes are governed by the following model equations:     Table 1.  Basic reproduction number. To understand the transmission dynamics, we analyzed the basic reproduction number 0  , which is defined as the number of secondary cases produced by a single infective individual during his/her infectious period in an otherwise wholly susceptible population 23     .  0 increases when the sexual transmission route is included. 0  is increasing in a, b, c, η, m, β, κ, τ, θ, ν v , and decreasing in ν h ,γ h1 ,γ h2 ,μ v . In particular, β (affected by safe-sex education), m (affected by vector control), and a (affected by bed net and insect repellent) are controllable parameters. Note that β is proportional to the fraction of unprotected sexual contact.
Plug-and-play inference framework. The monthly simulated cases are obtained as where ρ denotes reporting ratio. We assume that the observed monthly cases (both confirmed and suspected) C i is a random sample from a Negative-binomial (NB) distribution ∼ = = where θ 1 denotes the parameter vector and l i is the density associated with C i and Z i . We apply the framework to three countries -Brazil, Colombia and El Salvador, which reported most cases continuously, and we denote the overall likelihood for the three countries as = ∏ . = L L j j overall 1 3