# A New Stochastic Model for the _Aedes aegypti_ Life Cycle and the Dengue Virus Transmission
## 2 Materials and Methods  
### Mathematical Model  
We develop a model that considers the arrival of infected mosquitoes in a place without infected humans as the initial condition and in addition to the bite transmission we also add two very important factors: (1) the vertical and venereal transmissions of the virus among mosquitoes that were reported experimentally, and (2) the biological stages, in detail, of the life cycle of the mosquito. The life cycle of the mosquito consists mainly of four stages: egg, larva (produced during four instars), pupa, and adult mosquito (adult mosquito refers to the mosquito that emerges from the pupa, i.e., the mosquito in the last stage. We will from now refer to an adult mosquito simply as a mosquito). The first three stages give rise to the so-called aquatic phase. Interestingly, only the fed pregnant female looks for a place to lay eggs, i.e., migration is possible due to the necessity for a breeding site since the oviposition is carried out on the surfaces of the breeding source such as tree holes, or any water storage container. Thereafter, each egg hatches, generating a larva, and the larva lives in the aquatic phase until it emerges as a mosquito. It is important to mention that in this latter stage begins the new reproductive cycle as well as the venereal and vertical transmissions of the virus. According to the experimental results of [PERSON] et al. [28], we consider that from the infected females oviposition a total of 55-68.6% of the larvae are infected and the VNT rate from male mosquitoes to virgin uninfected female is 31.6%. Once the eggs are hatched we consider the survival of the aquatic phase a ratio of male:female of 1.05:1.0 [30]. Another important point is that we consider as susceptible to infection only the mature virgin females population, in accordance to experimental evidence [28]. Finally, the transmission of the virus between the female mosquito and the human could be through the female mosquito bite. Figure 1 shows the cycles and the ways in which mosquitoes can contract or transmit the dengue virus. The main contributions of the model that we are presenting are: (i) all states for specific phases of the mosquito's life, (ii) all the interactions of mosquito-mosquito and mosquito-human as well as (iii) the evolution of the population in each state. For the sake of simplicity, we will here describe a few states and their interactions: the mature virgin states of females and males are denoted by \(S_{mvfm}\) and \(S_{mvm}\), respectively. Their populations are represented by \(n_{mvfm}\) and \(n_{mvm}\). Once they copulate, state \(S_{mvfm-mvm}\), they transition to the new states \(S_{fm}\), \(S_{m}\), or \(S_{ffm}\), which denote mature females and males that have lost their virginity and females who became pregnant because of the copulation, respectively. \(n_{mvfm-mvm}\), \(n_{fm}\), \(n_{m}\), and \(n_{ffm}\) specify the population in each state. The transitions occur as follows:  
\[n_{mvfm}+n_{mvm}\qquad\rightharpoonup\qquad \begin{array}{c}n_{mvfm-mvm},\\ k_{k}\\ \end{array} \]  
\[n_{mvfm-mvm}\qquad\rightharpoonup\qquad \begin{array}{c}n_{ffm}+n_{m},\\ k_{k}\\ \end{array} \]  
\[n_{mvfm-mvm}\qquad\rightharpoonup\qquad \begin{array}{c}n_{fm}+n_{m}.\\ k_{k}\\ \end{array} \]  
Figure 1: Scheme of the mosquito life stages and the virus transmission routes. The red boxes stand for the infected populations and the blue boxes stand for the uninfected populations. The dashed arrows represent the virus infection routes and the solid arrows represent the healthy population’s routes or the recovered human population. In parenthesis we show the states related to each population. where \(k_{c}\), \(k_{dc}\) represent the transition rates and \(p_{p}\) is the probability of getting pregnant.