# High-Performance Computing and ABMS for High-Resolution COVID-19 Spreading Simulation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Agent Based Modelling and Simulation

^{2}) due to the need to identify the new agents at each simulation step. This is a big problem for applications running on a cluster of computers because, like this type of solution, execution is only effective if the processors spend most of their time processing rather than communicating [32]. Furthermore, it is often necessary to maintain the same order of execution of the agents to ensure the repeatability of the results, but in a multi-node system, this involves the use of synchronization protocols that at least increase execution times [33]. Finally, in such applications, the various agent’s creations may not be completely simple. In fact, it is possible to have sets of heterogeneous agents which probably require a different method of creation; it may be necessary to configure agents of the same type with different or random data, etc.

## 3. Agent Based Modeling and Simulation for Epidemic Scenarios

## 4. Actor Based Modeling and Simulation

#### 4.1. Using Actors for Large Scale ABMS Applications

#### 4.2. ActoDeS

#### 4.3. Using ActoDeS for ABMS Large Scale Applications

## 5. The Simulator

- Unique identifier
- Province or municipality of residence
- Age
- Number of daily contacts
- Current infection state
- If the individual is an essential worker
- If the individual wears a protective mask during the simulation period.

#### 5.1. Software Architecture with HPC

#### 5.2. Commuting Model

- ▪
- 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.

- ${c}_{i,j}$ is the probability that an individual living in i works or studies in municipality j
- ${N}_{i},{N}_{j}$ number of individuals living in municipality i (or j)
- $\mathsf{\theta}$ proportional constant equal to 0.0005
- $\mathsf{\tau}f$ inhabitants damping constant of i equal to 0.28
- $\mathsf{\tau}t$ inhabitants damping constant of j which changes according to the number of people living in j:
- 0.65 if the number of inhabitants is greater than 150,000
- 0.66 if the number of inhabitants is between 5000 and 150,000
- 0.78 if the number of inhabitants is less than 5000

- ${d}_{i,j}$ distance between the municipalities i and j
- $\mathsf{\rho}$ constant amplifying the dependence from distance which changes according to the number of people living in j:
- 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
- 2.5 0.78 if the number of inhabitants is less than 5000

- ${p}_{i,j}$ is the probability that an individual living in i works or studies in municipality j
- ${p}_{i}$ initial commuting probability of municipality i. This parameter can assume three different values according to the number of people who live in i:
- if the number of inhabitants is greater than 150,000
- 0.3 if the number of inhabitants is between 5000 and 150,000
- 0.4 if the number of inhabitants is less than 5000

- ${N}_{i},{N}_{j}$ number of individuals living in municipality i (or j)
- ${S}_{i,j}$ population that lives within the circle with radius equal to the distance between i and j minus the population living in i and j

## 6. Use-Cases: Modeling the Emilia-Romagna and Lombardy Regions

#### 6.1. Social-Demographic Model

- Single with children
- Single without children
- Single with children plus another adult
- Couple without children
- Couple without children plus another adult
- Couple with children
- Couple with children plus another adult
- Adults that live together
- Family groups (with at least one child)

- Any family group must contain at least one adult
- The age of each child must be between 18 and 43 years less than the younger parent
- The age difference of a couple is less than or equal to 15 years and they must be adult

- Kindergarten, attendance rate: 90%
- Preschool, attendance rate: 90%
- Elementary school, attendance rate: 100%
- Middle school, attendance rate: 100%
- High school, attendance rate: 92%
- University, attendance rate: 31%

- Kindergarten: 40 children
- Preschool: 20 children
- Elementary school: 19 children
- Middle school: 21 lads
- High school: 21 lads
- University: 34 lads

- 15–19 years: 8%
- 20–26 years: 30%
- 27–34 years: 62.5%
- 35–54 years: 73.5%
- 55–70 years: 54.3%

- Very small company: up to 5 employees
- Small company: up to 9 employees
- Small-medium company: up to 19 employees
- Medium company: up to 49 employees
- Medium-large company: up to 99 employees
- Large company: up to 249 employees
- Very large company: over 250 employees

#### 6.2. Restrictions

- White: no restrictions
- White from 10/18 to 10/24: represents the restrictions introduced on 18 October 2020: no limitations regarding work, but high schools and universities at 50% in attendance. The number of daily interactions is reduced by 40% to shape the closure of businesses such as bars after 9 pm and restaurants after midnight.
- White from 10/25 to 11/05: represents the latest restrictions introduced before the zoning: no limitations regarding work, but schools and universities still at 50% in attendance. Closing of activities such as gyms, theaters and cinemas, closing restaurants after 6 pm, also the recommendation not to move. The number of social interactions is reduced by 50% compared to the initial value.
- Yellow: high schools and universities closed (100% distance learning). Curfew from 22.00. Closure to the public of exhibitions, museums and other places of culture such as archives and libraries. To model these additional restrictions, the number of social interactions is reduced by 60%.
- Orange: in addition to the limitations of the yellow area, the prohibitions on moving between municipalities except for proven work needs and the closure of catering activities (excluding take-away) are added. The use of smart working is encouraged and recommended even for workers, excluding essentials. In the model, social interactions are reduced by 70% compared to the initial values and cannot take place outside the municipalities (except those related to work activities).
- Red: in addition to the limitations of the orange zone, the prohibition of movement within the municipality. For this, the daily interactions are reduced by 80% and reduced to zero those regarded as usual ones.

#### 6.3. Spreading Parameters’ Selection

#### 6.4. Results for Emilia-Romagna

#### 6.5. Results for the Lombardy Region

## 7. Discussion

- The contagion susceptibility by age
- The protection achieved thanks to wearable protective devices
- The quarantine mechanism

_{0}. The third one [67] (SEIR 3 in Figure 16) optimizes the model’s parameters using computational swarm intelligence to model the early stage of the pandemic in Italy. Moreover, we presented in Table 3 a numerical analysis of these models compared with ours. The chart and the analysis showed that our model produces a more accurate prediction of the infection spreading in terms of RMSE and the Pearson correlation.

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**The simulator’s architecture on the High Performance Computing (HPC) architecture divides individuals in several actor spaces on different computational nodes.

**Figure 5.**Image representing the difference between the gravity model and the radial one. In this example, we consider the municipality of Albinea (departure) and the municipality of Reggio nell’Emilia (arrival).

**Figure 15.**Simulation results without damping parameters with different transmission probability (0.2, 0.4, 0.6, 0.8, 1) (incremental representation—y-axis scale is logarithmic).

**Figure 16.**Comparison among our model, the serological data, the real data and the SEIR models (incremental representation—y-axis scale is logarithmic).

**Table 1.**Pearson correlation and root mean squared error (RMSE) of our model with respect to the serological data projection.

RMSE | Pearson Correlation | |
---|---|---|

Simulated Data—Real Data | 398,042 | 0.9903 |

**Table 2.**Comparison of our model to the null models in terms of root mean squared error (RMSE) and Pearson correlation n.

RMSE | Pearson Correlation | |
---|---|---|

Simulated Data—Serological | 105,343 | 0.9903 |

Null model with TP = 0.2–Serological | 1,301,649 | 0.8674 |

Null model with TP = 0.4–Serological | 4,662,723 | 0.9456 |

Null model with TP = 0.6–Serological | 5,840,197 | 0.9805 |

Null model with TP = 0.8–Serological | 6,422,819 | 0.9763 |

Null model with TP = 1–Serological | 6,759,554 | 0.9616 |

**Table 3.**Comparison of our model to the SEIR models in terms of root mean squared error (RMSE) and the Pearson correlation.

RMSE | Pearson Correlation | |
---|---|---|

Simulated Data—Serological | 105,343 | 0.990 |

SEIR 1—Serological | 269,197 | 0.769 |

SEIR 2—Serological | 270,060 | 0.859 |

SEIR 3—Serological | 318,034 | 0.713 |

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**MDPI and ACS Style**

Pellegrino, M.; Lombardo, G.; Cagnoni, S.; Poggi, A.
High-Performance Computing and ABMS for High-Resolution COVID-19 Spreading Simulation. *Future Internet* **2022**, *14*, 83.
https://doi.org/10.3390/fi14030083

**AMA Style**

Pellegrino M, Lombardo G, Cagnoni S, Poggi A.
High-Performance Computing and ABMS for High-Resolution COVID-19 Spreading Simulation. *Future Internet*. 2022; 14(3):83.
https://doi.org/10.3390/fi14030083

**Chicago/Turabian Style**

Pellegrino, Mattia, Gianfranco Lombardo, Stefano Cagnoni, and Agostino Poggi.
2022. "High-Performance Computing and ABMS for High-Resolution COVID-19 Spreading Simulation" *Future Internet* 14, no. 3: 83.
https://doi.org/10.3390/fi14030083