# A New Stochastic Model for the _Aedes aegypti_ Life Cycle and the Dengue Virus Transmission
## 2 Materials and Methods
### Mathematical Model
Let us take the interaction between a fertilized female (\(S_{ffm}\)) and an uninfected human (\(S_{nih}\)); after the interaction we get the new states \(S_{ffmf}\) and \(S_{nih}\), this is:  
\[n_{ffm}+n_{nih} \underset{k_{b}}{\rightharpoonup} n_{ffm-nih},\] \[n_{ffm-nih} \underset{k_{rb}}{\rightharpoonup} n_{ffmf}+n_{nih},\]  
where \(n_{ffm}\), \(n_{nih}\), and \(n_{ffmf}\) represent the population of fertilized female, the uninfected human, and the fed fertilized female after biting an uninfected human. \(k_{b}\) and \(k_{rb}\) represent the transition rates. Once a transition occurs, the population in each state is updated. In Tables 1-3 we show all the events: the state and the population of uninfected (Table 1), infected (Table 2), and bearer populations that are not contagious yet (Table 3). In Tables 4-6 we specify the transition rates among states and the effect on the populations involved. Tables 7 and 8 show all the parameter values used in the numerical simulations. It is important to clarify that the mature virgin male population is considered as part of the mature male population; the mature virgin infected female population is considered part of the mature infected female population and the mature virgin infected male population is considered part of the mature infected male population. However, it is important to consider the population of the mature virgin female as an independent population because it is the population susceptible to contract the virus in a venereal way. See Figures 2-4 for a better understanding of the transitions among all the possible states described in Figure 1. \begin{table}
\begin{tabular}{l l l l} \hline \hline  & **Event** & **State** & **Population** \\ \hline
1 & Virgin female emergence & \(S_{pfm}\) & \(n_{pfm}\) \\
2 & Virgin male emergence & \(S_{vm}\) & \(n_{vm}\) \\
3 & Mature virgin female & \(S_{mvfm}\) & \(n_{mv/m}\) \\
4 & Mature virgin female and male mating & \(S_{mvfm-m}\) & \(n_{mv/m-mvm}\) \\
5 & Mature female & \(S_{fm}\) & \(n_{fm}\) \\
6 & Mature male & \(S_{m}\) & \(n_{m}\) \\
7 & Mature female and male mating & \(S_{fm-m}\) & \(n_{fm-m}\) \\
8 & Fertilized female & \(S_{ffm}\) & \(n_{ffm}\) \\
9 & Uninfected human & \(S_{nih}\) & \(n_{nih}\) \\
10 & Recovered human & \(S_{th}\) & \(n_{rh}\) \\
11 & Fertilized female bites a uninfected human & \(S_{ffm-nih}\) & \(n_{ffm-nih}\) \\
12 & Fertilized female bites a recovered human & \(S_{ffm-th}\) & \(n_{ffm-th}\) \\
13 & Fertilized female and fed & \(S_{ffmf}\) & \(n_{ffmf}\) \\
14 & Eggs from uninfected female & \(S_{enifm}\) & \(n_{enifm}\) \\
15 & Uninfected larva & \(S_{l}\) & \(n_{l}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Representation of the states and populations in the life cycle of the uninfected mosquitoes and their interaction with uninfected and recovered humans. The mature virgin male population and the mature male population were considered as one as both events do not play a role in the spread of the virus. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Transition Rate** & **Event** & **Transition Rate** & **Event** \\ \hline \(k_{hh}\) & \(n_{nih}\to n_{nih}+1\) & \(k_{ovi}\,n_{ffmf}\) & \(n_{enifm}\to n_{enifm}+\overline{n_{eggso}}\) \\  & & \(n_{fm}\to n_{fm}+1\) \\  & & \(n_{ffm}\to n_{ffm}-1\) \\  & & \(n_{ffm}\to n_{ffm}-1\) \\ \hline \(\mu_{e}\,n_{enifm}\) & \(n_{enifm}\to n_{enifm}-1\) & \(k_{el}\,n_{enifm}\) & \(n_{l}\to n_{l}+1\) \\  & & \(n_{enifm}\to n_{enifm}-1\) \\ \hline \(\mu_{a}\,n_{l}+(f_{BS}\,\alpha\,n_{l}\,(n_{l}-1))\) & \(n_{l}\to n_{l}-1\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Transition rates and the effect on the population considered in the development in the life cycle of the uninfected mosquito and its interaction with uninfected humans.