Impact of weather seasonality and sexual transmission on the spread of Zika fever

We establish a compartmental model to study the transmission of Zika virus disease including spread through sexual contacts and the role of asymptomatic carriers. To incorporate the impact of the seasonality of weather on the spread of Zika, we apply a nonautonomous model with time-dependent mosquito birth rate and biting rate, which allows us to explain the differing outcome of the epidemic in different countries of South America: using Latin Hypercube Sampling for fitting, we were able to reproduce the different outcomes of the disease in various countries. Sensitivity analysis shows that, although the most important factors in Zika transmission are the birth rate of mosquitoes and the transmission rate from mosquitoes to humans, spread through sexual contacts also highly contributes to the transmission of Zika virus: our study suggests that the practice of safe sex among those who have possibly contracted the disease, can significantly reduce the number of Zika cases.


Supplementary information S.1 The governing equations
In accordance with the transmission diagram in Figure 3 and the parameter description given in Table 1, introducing the notation N h (t) and N f (t), for the total human and total female human population, respectively, the corresponding system of differential equations takes the form Here, B h and d denote human birth and death rate, respectively, β stands for transmission rate from symptomatically infected men to susceptible women, while for transmission rates from exposed, asymptotically infected and convalescent men to women are obtained by multiplying β by κ e , κ a and κ r , respectively. The parameter θ is the fraction of asymptomatically infected among all infected people. The length of latent period for humans is 1/ν h and 1/γ a , 1/γ s denote the length of infected period for asymptotically and symptomatically infected people, respectively, while 1/γ r is the length of the period during which recovered men are still infectious through sexual contact. The functionsα h (t),α v (t) andB v (t) denote transmission rate from an infectious mosquito to a susceptible human, transmission rate from an infected human to a susceptible mosquito and mosquito birth rate, respectively. These functions are assumed to be time periodic with one year as a period and, following e.g.
[49] they are assumed to be of the formα where P is period length, a, b are free adjustment parameters and α h , α v , B v are constants. Just like in the case of human-to-human transmission, we also introduce the modification parameters η e , η a for infectiousness of exposed and asymptomatically infected people, respectively. We have 1/ν v for the length of the latent period for mosquitoes, while average life span of mosquitoes is given by 1/µ.
The introduction of a nonautonomous model was needed to reproduce an epidemic with multiple peaks; this is supported by Figure S.1 where we tried to fit the autonomous model obtained from (S.1) by setting the time-dependent parameters (mosquito birth and death rates and biting rates) constant.

S.2 Derivation of the basic reproduction number of the autonomous model
To calculate the basic reproduction number R 0 of the autonomous model obtained from (S.1) by setting the time-dependent parameters (mosquito birth and death rates and biting rates) constant, we follow the general approach established in [50]. Given the infectious states E f , I a f , I s f , E m , I a m , I s m , I r m , E v and I v in (S.1), we can create the transmission vector F representing the new infections flowing only into the exposed compartments given by while the transition vector V which denotes the outflow from the infectious compartments in (S.1), is given by here we note that the notations α h , α v and B v stand for the (now constant) transmission rate from an infectious mosquito to a susceptible human, transmission rate from an infected human to a susceptible mosquito and mosquito birth rate, respectively. Substituting the values in the disease-free equilibrium N h = B h d and N f = B h 2d , we compute the Jacobian F from F given by and the Jacobian V from V given by therefore the characteristic polynomial of the next generation matrix F V −1 is The characteristic polynomial therefore is the following cubic equation given by According to [50], the basic reproduction number R 0 is the spectral radius of F V −1 . Thus, the basic reproduction number R 0 corresponds to the dominant eigenvalue given by the root of the cubic equation

S.3 Numerical estimation of the basic reproduction number of the nonautonomous model
In periodic epidemiological models, one can determine the basic reproduction as the spectral radius of a linear integral operator on a space of periodic functions (see [39] for details). The value of the basic reproduction number generally cannot be calculated analytically, but one can numerically approximate it. To do so, first, one writes the model where F i (t, x) stands for the be the rate of new infections in compartment i and x) denoting the rate of transfer into compartment i by all means different from new infections, and V − i (t, x) is the rate of transfer out of the ith compartment. Let m be the number of infectious compartments. The matrices F (t) and V (t) are defined as ∂x j 1≤i,j≤m and V (t) = ∂V i (t,x 0 (t)) ∂x j 1≤i,j≤m and consider the linear periodic equation with λ ∈ (0, ∞). First we find the monodromy matrix Φ of (S.2) by finding m linearly independent solutions, most simply by taking the (linearly independent) unit vectors of R m as initial values.