# High-Performance Computing and ABMS for High-Resolution COVID-19 Spreading Simulation
## 5 The Simulator
### Commuting Model  
To evaluate the robustness of our model and our hyper-parameters, we decided to add further detail to our model exploiting highly detailed data on socio-demographic structure in Italy and the commuting effects between provinces and municipalities. For high school students, university students and workers, there is the possibility that the employment place is located in a different town than that of residence, or even in a different province. Commuting is very common nowadays and plays an important role in the infection spreading even in the most isolated municipalities. Therefore, it is essential to model this phenomenon in a way that is as closely comparable to reality as possible. Commuting models commonly rely on the effectiveness of two parameters:  
* The grain: the finer the grain of the model, the more accurate the commuting model is. * The economic quotient: the most effective models take into account the economic quotient of every zone. Moreover, a municipality with a high number of businesses attracts more workers than another with few job opportunities. After some analyses, we decided to use the average results between a gravitational model that is presented in [5] and the radial one that is explained in [4]. We chose this approach because the first model has a better commuters' distribution in the various municipalities near the departure one. On the other hand, the second model computes in a more reliable way the correct number of people moving from a given municipality, but tends to distribute the contacts only to the closest town. The gravity model puts in a relationship the inhabitant number of the departure municipality, the inhabitant number of the arrival municipality, and the Euclidean distance between them:  
\[c_{ij}=\theta\frac{N_{i}^{\text{\tiny T}}/N_{j}^{\text{\tiny T}}}{d_{ij}^{ \text{\tiny P}}} \tag{1}\]  
Figure 4: The simulation process algorithm. where:  
1. \(c_{i,j}\) is the probability that an individual living in \(i\) works or studies in municipality \(j\)
2. \(N_{i}\), \(N_{j}\) number of individuals living in municipality \(i\) (or \(j\))
3. \(\theta\) proportional constant equal to 0.0005
4. \(\tau f\) inhabitants damping constant of \(i\) equal to 0.28
5. \(\tau t\) inhabitants damping constant of \(j\) which changes according to the number of people living in \(j\): 1. 0.65 if the number of inhabitants is greater than 150,000
2. 0.66 if the number of inhabitants is between 5000 and 150,000
3. 0.78 if the number of inhabitants is less than 5000
6. \(d_{i,j}\) distance between the municipalities \(i\) and \(j\)
7. \(\rho\) constant amplifying the dependence from distance which changes according to the number of people living in \(j\): 1. 3.05 if the number of inhabitants is greater than 150,000 2.95 if the number of inhabitants is between 5000 and 150,000
3. 2.5 0.78 if the number of inhabitants is less than 5000  
Due to the lack of data of an economic quotient, we supposed that a municipality with many inhabitants also has a higher economic attraction. For this reason, the constants are set according to the number of people living in the municipality. The radial model, on the other hand, puts in a relationship the inhabitant number of the departure municipality, the inhabitant number of the arrival municipality, and the number of people living in the circle with a radius equal to the distance between the two municipalities. Hence:  
\[p_{i,j}=p_{i}\frac{N_{i}N_{j}}{\left(N_{i}+S_{i,j}\right)\left(N_{i}+N_{j}+S_ {i,j}\right)} \tag{2}\]  
where:  
1. \(p_{i,j}\) is the probability that an individual living in \(i\) works or studies in municipality \(j\)
2. \(p_{i}\) initial commuting probability of municipality \(i\). This parameter can assume three different values according to the number of people who live in \(i\): 1. if the number of inhabitants is greater than 150,000 2. 0.3 if the number of inhabitants is between 5000 and 150,000 3. 0.4 if the number of inhabitants is less than 5000
3.