# Evacuation Management System for Major Disasters
## 2 Mathematical Model
### Evacuation Time Estimation  
Once the evacuation routes are generated, the next step is simulating the evacuation process (pedestrian + vehicular). As mentioned in the introduction section, the challenge consists of performing real-time simulations while maintaining granularity and accuracy in the models. To address this, we used Monte Carlo simulations to generate representative and significant samples within a short period of time. These real-time simulations also help to explore the results of a current or an expected evacuation strategy, e.g., detecting in advance the potential conflicts between routes (e.g., congestions) and whether the distribution of shelters and assembly points is appropriate. _Pedestrian model_ calculates the movement times of individuals from their starting locations (e.g., households) to the assembly points. Each pedestrian (\(p\)) is modelled using three random variables: (1) initial geographic location (\(l_{p}\)), (2) pre-evacuation time (\(t_{prep}\)) and (3) walking speed (\(v_{p}\)). These variables are assigned according to the distributions (Table 1) that have been widely studied in the literature [29; 30; 31]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Var** & **Distribution** & **Details** \\ \hline \multirow{4}{*}{\(l_{p}\)} & & \(\phi=u_{1}\cdot|\phi_{max}-\phi_{min}|\) & [Lng] \\  & Uniform. & \(\theta=u_{2}\cdot|\phi_{max}-\phi_{min}|\) & [Lat] \\  & & \(L_{2}-Euc\). \(Polygon\) & \\  & & \(\mathsf{p}=(\phi,\theta)\cap L_{2}\) \\ \hline \multirow{2}{*}{\(v_{p}\)} & \multirow{2}{*}{Normal.} & \(\mathsf{r}=\sqrt{-2\cdot Ln(U_{1})}\cdot Sin(2\pi U_{2})\) \\  & & \(v_{p}=\mu_{v}+\sigma_{v}\cdot r\) \\ \hline \multirow{2}{*}{\(t_{prep}\)} & \multirow{2}{*}{Log-Normal} & \(\mathsf{r}=\sqrt{-2\cdot Ln(U_{1})}\cdot Sin(2\pi U_{2})\) \\  & & \(t_{prep}=e^{t_{prep}+\sigma_{prep}\cdot r}\) \\ \hline \multirow{2}{*}{\(U_{1}\sim lR(0,1)\)} & \(U_{2}\sim lR(0,1)\) & \(U_{2}\sim lR(0,1)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Pedestrian parameters estimation. Figure 3: Evacuation route representation from evacuation areas (blue polygons) avoiding damaged areas (red polygons). Hence the movement time for each pedestrian \(t_{rp}\) is calculated as follows:  
\[t_{rp}=t_{prep}+\frac{\min\{Dist\{l_{p},C_{i}\}\}}{v_{p}}\]  
where the function \(Dist(l_{p},C_{i})\) is the distance between the starting location of the evacuee and the assembly point \(C_{i}\). The default population distribution is taken from WorldPop ([[https://www.worldpop.org/](https://www.worldpop.org/)]([https://www.worldpop.org/](https://www.worldpop.org/)), accessed on 25 July 2022), which contains project data that use both the integration of census data and aerial imagery via satellite in which people counts and density are provided with a resolution of 100 m\({}^{2}\). _Velicular model_ calculates for each route \(R_{i}\) the traffic density, average speed and number of evacuated people as a function of time. This model uses the pedestrian movement times, vehicles occupancy, boarding time and vehicle expected distribution as random inputs to simulate, via cellular automata, the current traffic status of the routes and the simulated vehicles they contain by following a microscopic modelling approach [32]. The calculated routes by the previous model are split into road sections with common characteristics. Road sections are produced by solving the graph problem, erasing duplicated instances that are used by more than one route at the same time, and calculating the vehicle interactions emanating from different routes that converge in the same road section and the distribution of vehicles when a road section is divided (Figure 4). The interaction between vehicles in the same road section is produced by three factors: (1) traffic density \(k\) (current), \(k_{c}\) (critical) and \(k_{j}\) (jam); (2) average speed \(u,~{}u_{c}\) and \(u_{j}\); and (3) route maximum density \(k_{max}\). These variables follow a particular function that was previously studied in [33].