# A New Stochastic Model for the _Aedes aegypti_ Life Cycle and the Dengue Virus Transmission
## 3 Results  
As the first step, in order to validate our stochastic model, we used the same initial conditions and parameter values as in [PERSON] et al. [20] in order to reproduce their results. We therefore set the infected population to zero. The initial conditions were: uninfected eggs number (5000), uninfected humans (1000), number of breeding sites (150), and the same function for the seasonal variation (denoted here by \(T_{a}\)) [\(T_{a}=a+b\) cos \(\left(\frac{2\pi t}{36525\,\text{days}}+c\right)\), \(a=18.0\)\({}^{\circ}\)C, \(b=6.7\)\({}^{\circ}\)C and \(c=9.2\)]. In this case zero-day correspond to the first of July. Despite 100 realizations (numerical simulations) being performed for each situation, from \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Probability** & **Representation** & **Value** & **Reference** \\ \hline Probability of being female & \(p_{fm}\) & 0.49505 & [30] \\ Probability of being male & \(p_{m}\) & 0.50495 & [30] \\ Infective biting probability & \(k_{ib}\) & 0.75 & [41] \\ Probability of being impregnated & \(p_{p}\) & 0.8 & [14] \\ Probability of virus dissemination (VT) & \(p_{v}\) & 0.68 & [28] \\ Probability of virus venereal transmission (VNT) & \(pront\) & 0.316 & [28] \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Probabilities for (i) the sexual gender of the mosquito, (ii) to be fertilized, and (iii) for the transmission of the virus. Figure 5-8 we show 3 realizations for each population in order to observe the time series profiles in a better way. Figure 5a,b,d correspond to the female mosquito, male mosquito, and larva populations, respectively, whereas Figure 5c shows the egg population, which is in accordance with the evolution profile previously reported by [PERSON] et al. [20], i.e., a fast growth of the egg population in spring and its abrupt reduction in autumn. [PERSON] et al. [43] concluded that the mosquito dispersal is an advantageous persistence strategy. The authors studied a case in which females travel looking for the availability of oviposition sites; however, they also mention that wind, ships, trucks, or the public transport are mechanisms for long distance dispersal of the _Aedes aegypti_. [PERSON] et al. [44] have recently discussed the origin of the domestic form and the dispersal and evolution of the _Aedes aegypti_ in different continents. Based on the line of the dispersion of mosquitoes and by considering the mosquito longevity range to be from 20 to 140 days in the natural environment [13; 15; 45], and also considering that the human viremic period range is from 3 to 7 days [22; 41], we simulated a scenario in which 5 infected females (to ensure a sufficient population of eggs in our model) arrive to oviposit in a place where there is a susceptible human population but without any infected human as the initial condition, in order to study the virus propagation on this population. In this context, our second case of study was to consider that the virus can only be transmitted between infected hosts through biting. As mentioned above, the initial conditions were set to five infected females. It is important to clarify that from now on  
Figure 5: Time series of 10 realizations (in different colours) with the same initial and environmental conditions used by [PERSON] et al. [20]. (**a**) shows the female population, (**b**) shows the male population, (**c**) shows the eggs population, and (**d**) shows the larva population. The time series correspond to one year. \(T_{a}=a+b\cos\left(\frac{2\pi t}{365.25\,\text{days}}+c\right)\), \(a=18.0\)\({}^{\circ}\)C, \(b=6.7\)\({}^{\circ}\)C and \(c=9.2\), \(BS=150\) and \(n_{enifm}=5000\). we will consider the temperature reported for Tuxtla Gutierrez city, from 1 January 2013 to 31 December 2016, in our analysis. The results of three realizations are presented in Figure 6. Figure 6a-f show the time series of the populations of females, males, eggs, larvae, infected females, and infected humans, respectively.