# A New Stochastic Model for the _Aedes aegypti_ Life Cycle and the Dengue Virus Transmission
## 2 Materials and Methods
### Mathematical Model
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Transition Rate** & **Event** & **Transition Rate** & **Event** \\ \hline \(k_{t}\,n_{l}\,p_{fm}\) & \(n_{vfm}\to n_{vfm}+1\) & \(k_{t}\,n_{l}\,p_{m}\) & \(n_{vfm}\to n_{vm}+1\) \\  & \(n_{l}\to n_{l}-1\) & & \(n_{l}\to n_{l}-1\) \\ \hline \(k_{m}\,n_{vm}\) & \(n_{m}\to n_{m}+1\) & \(k_{m}\,n_{vfm}\) & \(n_{mfm}\to n_{nvfm}+1\) \\  & \(n_{vm}\to n_{vm}-1\) & & \(n_{vfm}\to n_{vfm}-1\) \\ \hline \(k_{c}\,n_{m}\,n_{mvfm}\) & \(n_{mvfm}\to n_{mvfm-m}+1\) & \(k_{c}\,n_{m}\,n_{fm}\) & \(n_{fm-m}\to n_{fm-m}+1\) \\  & \(n_{mvfm}\to n_{mvfm}-1\) & & \(n_{fm}\to n_{fm}-1\) \\  & \(n_{mvfm}\to n_{mvfm}-1\) & & \(n_{fm}\to n_{fm}-1\) \\  & \(n_{m}\to n_{m}-1\) & & \(n_{m}\to n_{m}-1\) \\ \hline \(k_{dc}\,n_{mvfm-m}\,p_{p}\) & \(n_{mvfm-m}\to n_{mvfm-m}-1\) & \(k_{dc}\,n_{mvfm-m}\,(1-p_{p})\) & \(n_{mvfm-m}\to n_{mvfm-m}-1\) \\  & \(n_{ffm}\to n_{ffm}+1\) & & \(n_{fm}\to n_{fm}+1\) \\  & \(n_{m}\to n_{m}+1\) & & \(n_{m}\to n_{m}+1\) \\ \hline \(k_{dc}\,n_{fm-m}\,p_{p}\) & \(n_{fm-m}\to n_{fm-m}-1\) & \(k_{dc}\,n_{fm-m}\,(1-p_{p})\) & \(n_{fm-m}\to n_{fm-m}-1\) \\  & \(n_{ffm}\to n_{ffm}+1\) & & \(n_{fm}\to n_{fm}+1\) \\ \hline \(k_{dc}\,n_{fm-m}\,p_{p}\) & \(n_{fm-m}\to n_{fm-m}-1\) & \(k_{dc}\,n_{fm-m}\,(1-p_{p})\) & \(n_{fm-m}\to n_{fm-m}-1\) \\  & \(n_{ffm}\to n_{ffm}+1\) & & \(n_{fm}\to n_{fm}+1\) \\  & \(n_{m}\to n_{m}+1\) & & \(n_{m}\to n_{m}+1\) \\ \hline \(k_{b}\,n_{ffm}\,n_{nih}\) & \(n_{ffm-nih}\to n_{ffm-nih}+1\) & \(k_{rb}\,n_{ffm-nih}\) & \(n_{ffm-nih}\to n_{ffm-nih}-1\) \\  & \(n_{nih}\to n_{nih}-1\) & & \(n_{nih}\to n_{nih}+1\) \\  & \(n_{ffm}\to n_{ffm}-1\) & & \(n_{ffm}\to n_{ffm}+1\) \\ \hline \(k_{b}\,n_{ffm}\,n_{nih}\) & \(n_{ffm-nih}\to n_{ffm-nih}+1\) & \(k_{rb}\,n_{ffm-nih}\) & \(n_{ffm-nih}\to n_{ffm-nih}-1\) \\  & \(n_{nih}\to n_{nih}-1\) & & \(n_{nih}\to n_{nih}+1\) \\  & \(n_{ffm}\to n_{ffm}-1\) & & \(n_{ffm}\to n_{ffm}+1\) \\ \hline \((\mu_{T}+\mu_{n})\,n_{vm}\) & \(n_{vm/m}\to n_{vmfm}-1\) & \((\mu_{T}+\mu_{n})\,n_{mm}\) & \(n_{vm}\to n_{mm}-1\) \\ \hline \((\mu_{T}+\mu_{n})\,n_{fm}\) & \(n_{fm}\to n_{fm}-1\) & \((\mu_{T}+\mu_{n})\,n_{mm}\) & \(n_{mm}\to n_{mm}-1\) \\ \hline \((\mu_{T}+\mu_{n})\,n_{fm}\) & \(n_{fm}\to n_{fm}-1\) & \((\mu_{T}+\mu_{n})\,n_{m}\) & \(n_{m}\to n_{m}-1\) \\ \hline \((\mu_{T}+\mu_{n})\,n_{ffm}\) & \(n_{fm}\to n_{ffm}-1\) & \((\mu_{T}+\mu_{n})\,n_{fm}\) & \(n_{fm}\to n_{fm}-1\) \\ \hline \((\mu_{T}+\mu_{n})\,n_{ffm}\) & \(n_{fm}\to n_{ffm}-1\) & \((\mu_{T}+\mu_{n})\,n_{fm}\) & \(n_{fm}\to n_{fm}-1\) \\ \hline \end{tabular}
\end{table} 
Table 4: \(Cont.\)  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Transition Rate** & **Event** & **Transition Rate** & **Event** \\ \hline \(k_{ovi}\,n_{ffmf}\,(1-p_{vt})\) & \(n_{enifm}\to n_{enifm}+\overline{n_{ggo}}\) & \(k_{ovi}\,n_{ffmf}\,p_{vt}\) & \(n_{eifm}\to n_{enifm}+0.67\,\overline{n_{ggo}}\) \\  & \(n_{ifm}\to n_{ifm}+1\) & & \(n_{eifm}\to n_{eifm}+0.33\,\overline{n_{ggo}}\) \\  & \(n_{ffmf}\to n_{ffinf}-1\) & & \(n_{fifmf}\to n_{ffmf}-1\) \\ \hline \(k_{el}\,n_{  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Transition Rate** & **Event** & **Transition Rate** & **Event** \\ \hline \(k_{dc}\,n_{v/fm-m}\,p_{p}\) & \(n_{v/fm-m}\,n_{v/fm-m}\,n_{v/fm-m}\,1\) & \(k_{dc}\,n_{v/fm-m}\,(1-p_{p})\) & \(n_{v/fm-m}\,n_{v/fm-m}\,-1\) \\  & \(n_{fv/fm}\,n_{fv/fm}\,n_{fv/fm}\,+1\) & & \(n_{v/fm}\,n_{v/fm}\,n_{v/fm}\,+1\) \\  & \(n_{m}\,n_{m}\,+1\) & & \(n_{  
To study the evolution of the populations we implemented the numerical analysis in Python using the Gillespie algorithm [31,32]. One hundred realizations (numerical simulations) were performed for each scenario. The algorithm is described below:  
1. Set the initial conditions of each state (\(S_{i}\)), where \(i\) denotes the representation of each event (see Tables 1-3) and initialize the simulation time (\(t=t_{0}\)). 2. Calculate the transition rates and their sum (denoted by \(a_{0}\)). 3.