# SARS-CoV-2 Spread Forecast Dynamic Model Validation through Digital Twin Approach, Catalonia Case Study

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## Abstract

**:**

## 1. Introduction

_{t}). To measure the transmission potential of a disease, we use the basic reproduction number R

_{0}; this value represents the average number of secondary infections produced by a single infection in a population. The average number of secondary cases will be lower than the basic reproduction number because not all the population will be susceptible; this will be represented by the effective reproductive number (R

_{t}). If R

_{t}> 1, the number of cases will increase, on R = 1, remains stable, and with R < 1, the number of cases will decrease. We can calculate R

_{t}as ${R}_{t}={R}_{0}{P}_{s}$, being P

_{s}, the fraction of susceptible population. We also must consider the complexity to cope with the large amount of information generated by all the scientists working on SARS-CoV-2 along with the project evolution, information that must be discussed while building the model. The analysis of this phenomenon needs experts from different areas collaborating in the study using different viewpoints. Moreover, when experts make a model, they can introduce errors due to a wrong understanding of the system behavior. Also, the model implementation can introduce errors [18].

#### A Digital Twin Approach

## 2. Materials and Methods

#### 2.1. System Dynamics Model

#### 2.2. Python Model

_{0}) from the containment factor ($\rho $), the transmission rate (β), and the recovery rate (γ) with the Equation (7). The basic reproduction number, R

_{0}, see [29], represents the average number of secondary cases that result from the introduction of a single infectious case in a susceptible population during the infectiousness period. The effective reproduction number, Rt, is the same concept but after applying the containment measures:

#### 2.3. The SDL Model

## 3. Results

#### 3.1. Models Coding and Calibration

#### 3.2. Second Wave Calibration

#### 3.3. Third-Wave Calibration

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. SDL

**Start.**This element defines the initial condition for a PROCESS diagram.

**State.**The PROCESS must always start in a STATE, and this owns a name.

**Input.**The PROCESS starts an execution when an INPUT receives the SIGNAL for this INPUT. All the STATES can own several different INPUTS to work with the different SIGNALS one can receive.

**Create.**This element allows the creation of an AGENT.

**Task.**To interpret a piece of code, we can use the TASK element. In our approach, we can use C on this element.

**Output.**To send a SIGNAL, we must use the OUTPUT element. We can also add parameters to the SIGNALS and describe the destination if ambiguity about the signal destination exists. We can direct the communication specifying destinations using a PROCESS identifier (PId), an identifier that must own all the PROCESS. Also, we can send using the sentence via path. We can use four PId expressions: (i)

**self**, an agent’s own identity; (ii)

**parent**, the agent that created the agent (Null for initial agents); (iii)

**offspring**, the most recent agent created by the agent; (iv)

**sender**, the agent that sent the last signal input (null before any signal received). Also, we can use {CUR_CELLS} and {ALL_CELL} to send the information to a specific cell of the CA.

**Decision**. To define a bifurcation, a decision point, we can use the DECISION.

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**Figure 3.**Fitting the changing points to detect regime shifts in the cumulative incidence curve (7 regimes in this picture).

**Figure 4.**SDL 2.5 model. Each color on the diagram represents a Compartment on the original SEIRD model (using the same colors). The exception is for the BContentionActions BLOCK that represents the different NPIs applied. Notice that BInfectiveDetected and BInfective refer to the single Infective compartment on the initial SEIRD model because on the SDL 2.5 version model, we can distinguish between real and detected cases. Notice that the 1(1) in the upper corner defines the number of pages that detail this diagram and the current page number.

**Figure 5.**SEIRD python-coded model results for Israel after the September 1 outbreak (top left) and for Catalonia before schools opening (September 14) (right). See the similarity in the patterns before the outbreak.

**Figure 6.**Forecast for the new infections (y-axis), the red line for the 2.5 SDL model (being this an optimistic scenario). We consider other scenarios in case 2.5 model hypotheses fail, model 2.6, and model 2.7.

**Figure 7.**Model 2.8 with the forecast for the second wave for the new cases (y-axis). Notice the divergence between the trend of the model and point A, the highest value of the data. We are presenting a scenario considering that the Christmas holidays have no acute effects on the β.

**Figure 9.**Model 2.9 now forecasts for new cases (y-axis) the end of the pandemic situation in Catalonia if the NPIs remain similar to the current (at 05 June 2021) and no new variants appear.

**Figure 10.**The different models that we develop due to the continuous validation process, new cases (y-axis).

Model Number | Description | Valid (at the Time of Writing This Paper) |
---|---|---|

1.9 | The initial model contains the initial growth and the total lockdown | No, the total lockdown was open. |

2.5 | Optimistical return to normality (schools and work). | No |

2.6 | Increase online learning and teleworking. | No |

2.7 | Pessimistic return to normality. | No |

2.8 | More NPIs application. | No |

2.9 | Readjusted the effect of the holidays and January restraints added. Adding the effects of the vaccination on the population. | Yes |

Id. | Code | Description | Population |
---|---|---|---|

6100 | LL | Lleida | 362,850 |

6200 | CT | Camp de Tarragona | 607,999 |

6300 | TE | Terres de l’Ebre | 176,817 |

6400 | GR | Girona | 861,753 |

6700 | CC | Catalunya Central | 526,959 |

7100 | AA | Vall d’Aran | 67,277 |

7801 | BS | Barcelona Sud | 1,370,709 |

7802 | BN | Barcelona Nord | 1,986,032 |

7803 | BC | Barcelona Ciutat | 1,693,449 |

All | CAT | Catalunya | 7,653,845 |

**Table 3.**Parameters for model 2.5 to be considered due to NPIs. % Det denotes real cases reporting level.

Event | Date (2020) | β | % Det | % Conf | NPIs |
---|---|---|---|---|---|

1 | 29 January | 1.2 | 0.1 | 0% | First infected |

2 | 08 Febrary | 1.2 | 0.25 | 0% | Initial tests |

3 | 15 March | 0.6 | 0.45 | 35% | Confinement |

4 | 23 March | 0.24 | 0.45 | 35% | Air space closes |

5 | 13 April | 0.2 | 0.45 | 25% | Workers partial comeback |

6 | 20 April | 0.18 | 0.45 | 25% | Free masks |

7 | 25 May | 0.18 | 0.45 | 25% | Phase 1 for some regions |

8 | 18 June | 0.18 | 0.54 | 0% | Phase 3 for BCN |

9 | 24 June | 1.2 | 0.54 | 0% | National day |

10 | 25 June | 0.18 | 0.54 | 0% | Phase 3 for BCN |

11 | 02 July | 0.3 | 0.54 | 0% | New normality |

12 | 17 July | 0.21 | 0.54 | 0% | Summer plateau |

13 | 15 September | 0.24 | 0.7 | 0% | School returns |

**Table 4.**β comparison. (*) In South Korea, the summer outbreak is so low that the model is not proper. (**) The forecast of a new outbreak with a β close to Israel proved to be accurate. (?) Information is not yet available, or the situation did not occur at the time of analysis of model 2.5.

Regime | Israel | S. Korea | Catalonia |
---|---|---|---|

First outbreak | 0.55 | 0.95 | 0.95 |

Lockdown | 0.12 | 0.09 | 0.15 |

Summer outbreak | 0.34 | (*) | 0.43 |

Summer plateau | 0.20 | (*) | 0.20 |

Reopening outbreak | 0.33 | 0.48 | 0.30 ^{(}**^{)} |

Partial lockdown | (?) | 0.12 | (?) |

**Table 5.**Parameters used on the 2.8 simulation scenarios: (1) as a result of the validation process, the correct parameter for this β is 0.3, being added on the 2.9 model. (2) the organization of the online courses needs a week to be fully implemented.

Event | Date | β | %Det | % Conf | Description Event |
---|---|---|---|---|---|

1 | 01 December 2019 | - | - | - | Start of simulation |

2 | 01 December 2019 | 0.81 | 0 | 0 | - |

3 | 11 December 2019 | 0.81 | 0.11 | 0 | Pandemic Beginning |

4 | 15 March 2020 | 0.81 | 0.17 | 0 | Confinement |

5 | 15 March 2020 | 0.81 | 0.17 | 0.35 | Confinement |

6 | 15 March 2020 | 0.25 | 0.17 | 0.35 | Confinement |

7 | 13 April 2020 | 0.25 | 0.17 | 0.2 | Workers partial comeback |

8 | 20 April 2020 | 0.16 | 0.17 | 0.2 | Free Masks |

9 | 06 May 2020 | 0.16 | 0.18 | 0.2 | Phase 1 for some regions |

10 | 01 June 2020 | 0.16 | 0.25 | 0.2 | Phase 3 for some regions |

11 | 18 June 2020 | 0.465 | 0.25 | 0.2 | Phase 3 for BCN |

12 | 18 June 2020 | 0.465 | 0.25 | 0 | Phase 3 for BCN |

13 | 22 June 2020 | 0.465 | 0.6 | 0 | New normality |

14 | 16 July 2020 | 0.21 | 0.6 | 0 | Summer plateau |

15 | 15 September 2020 | 0.34 | 0.6 | 0 | School returns |

16 | 20 October 2020 | 0.34 | 0.6 | 0.03 | University online ^{(2)} |

17 | 25 October 2020 | 0.34 | 0.6 | 0.1 | Movement and restaurants restrictions |

18 | 25 October 2020 | 0.15 | 0.6 | 0.1 | Movement and restaurants restrictions |

19 | 23 November 2020 | 0.3 | 0.6 | 0.1 | Reopening restaurants |

20 | 23 November 2020 | 0.3 | 0.6 | 0.03 | Reopening restaurants |

21 | 23 December 2020 | 0.21 (0.3) ^{(1)} | 0.6 | 0.03 | Holidays |

22 | 11 January 2021 | 0.3 | 0.6 | 0.03 | Schools Returns |

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**MDPI and ACS Style**

Fonseca i Casas, P.; Garcia i Subirana, J.; García i Carrasco, V.; Pi i Palomés, X. SARS-CoV-2 Spread Forecast Dynamic Model Validation through Digital Twin Approach, Catalonia Case Study. *Mathematics* **2021**, *9*, 1660.
https://doi.org/10.3390/math9141660

**AMA Style**

Fonseca i Casas P, Garcia i Subirana J, García i Carrasco V, Pi i Palomés X. SARS-CoV-2 Spread Forecast Dynamic Model Validation through Digital Twin Approach, Catalonia Case Study. *Mathematics*. 2021; 9(14):1660.
https://doi.org/10.3390/math9141660

**Chicago/Turabian Style**

Fonseca i Casas, Pau, Joan Garcia i Subirana, Víctor García i Carrasco, and Xavier Pi i Palomés. 2021. "SARS-CoV-2 Spread Forecast Dynamic Model Validation through Digital Twin Approach, Catalonia Case Study" *Mathematics* 9, no. 14: 1660.
https://doi.org/10.3390/math9141660