# Modeling the 2013 Zika Outbreak in French Polynesia: Intervention Strategies
## Methodology
### Mosquito Human Interaction (SEI-SEIR Model)  
The outbreak was simulated using agent-based models, where the human and adult mosquito population in the hosting stage are the agents that participate in the transmission of the disease. The total number of adult mosquitoes at the start of the experiment were assumed to be proportional to the size of the human population. The number of mosquitoes per human (\(m\)) has been shown to vary between two and four in previous studies [28; 29]. The base value for this parameter was set to three and reduced to study the impact of intervention strategies. This count includes both male and female adult populations. Since only female mosquitoes in the hosting stage are vectors of the disease, only these mosquitoes are treated as agents for this phase. As stated earlier, the transmission is controlled by two compartmental systems interacting with each other (Figure 1). The transition from one compartment to another is governed by the system of differential equations in Equations (2) and (3):  
\[ \begin{array}{rcl}\frac{\partial E_{H}}{\partial t}&=&-\beta_{VH}I_{V}\frac{ S_{H}}{N_{H}},\\ \frac{\partial E_{H}}{\partial t}&=&\beta_{VH}I_{V}\frac{S_{H}}{N_{H}}-\alpha_{ H}E_{H},\\ \frac{\partial I_{H}}{\partial t}&=&\alpha_{H}E_{H}-\gamma I_{H},\\ \frac{\partial R_{H}}{\partial t}&=&\gamma I_{H},\end{array}  \tag{2}\]\[ \begin{array}{rcl}\frac{\partial S_{V}}{\partial t}&=&\frac{\rho_{p}}{2}P+\rho_{A_ {b}}A_{a}-\beta_{HV}S_{V}\frac{I_{H}}{N_{H}}-(\mu_{A_{h}}+\rho_{A_{h}})S_{V},\\ \\ \frac{\partial E_{V}}{\partial t}&=&\beta_{HV}S_{V}\frac{I_{H}}{N_{H}}-(\mu_{A _{h}}+\rho_{A_{h}})E_{v}-\alpha_{V}E_{V},\\ \\ \frac{\partial I_{V}}{\partial t}&=&\alpha_{V}E_{V}-(\mu_{A_{h}}+\rho_{A_{h}})I _{V},\end{array}  \tag{3}\]  
The total human population on each island, \(N_{H}\), is divided into four compartments, Susceptibles (\(S_{H}\)), Exposed (\(E_{H}\)), Infectious (\(I_{H}\)) and Recovered (\(R_{H}\)), such that \(N_{H}=S_{H}+I_{H}+E_{H}+R_{H}\), and the vector population is divided into three compartments, Susceptibles (\(S_{V}\)), Exposed (\(E_{V}\)) and Infectious (\(I_{V}\)). The total vector population is dynamic as described in Section 3.1. Infectious vectors that bite susceptible humans transmit the disease with a probability of \(p_{ph}\) (probability of transmission from vectors to humans) and susceptible vectors that bite infected humans contract the disease with a probability of \(p_{htv}\) (probability of transmission from humans to vectors). The transmission coefficient for transmission from vector to human (\(\beta_{VH}\)) is defined as the product of the corresponding transmission probability and the bite rate (\(b_{i}\), number of bites per day) for infectious mosquitoes, similarly the transmission coefficient for transmission from human to vector (\(\beta_{HV}\)) is the product of the corresponding transmission probability and the bite rate (\(b_{s}\), number of bites per day) for susceptible mosquitoes. The transmission probabilities and the bite rates vary with temperature and the expressions derived for Dengue in [17] were used to calculate these parameters (Equation (4)). The bite rate for infectious mosquitoes is assumed to be higher (double) than the bite rate for susceptible mosquitoes in case of Dengue [30]. The same was assumed for Zika, \(b_{i}=2b_{s}\). An infected vector bites \(b_{i}\) humans every day, the bites corresponding to susceptible humans out of these \(b_{i}\) bites is equal to \(b_{i}\frac{S_{H}}{N_{H}}\) and the bites that successfully transmit the disease are \(p_{ph}b_{i}\frac{S_{H}}{N_{H}}\), therefore the number of humans that are infected each day is equal to \(\beta_{VH}I_{V}\frac{S_{H}}{N_{H}}\). The number of infected mosquitoes is calculated as \(\beta_{HV}I_{H}\frac{S_{V}}{N_{H}}\) similarly. The exposed individuals become infectious after \(\frac{1}{\alpha_{H}}\) days and the infectious individuals recover after \(\frac{1}{\gamma}\) days. Similarly the exposed mosquitoes become infectious in \(\frac{1}{\alpha_{V}}\) days.