# SEIR Modeling of the Italian Epidemic of SARS-CoV-2 Using Computational Swarm Intelligence

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Database

#### 2.2. Overview of the Generalized SEIR Model

^{−1}as a unit of measurement.

- β is called infection rate. It is the number of people that an infective person infects each day. It is equal to $p$, where b, or the contact rate, is the number of people an average person enters into contact with each day, and p is the probability that a contact provokes the transmission of the disease. In the SEIR model, β is the vector which transports people from the S category to the E category. It is multiplied by the ratio S/N to avoid counting contacts between two people who cannot infect each other (e.g., because one of them has already recovered, or because both are infective).
- γ is the inverse of the average latent time and governs the lag between having undergone an infectious contact and showing symptoms: in the equations, it brings people from the E category to the I category.
- λ and κ are the recovery rate and the death rate, respectively, and they are united together in a single parameter in the classical SEIR model. They give information about how fast the people may recover from the disease (1/λ is the average recovery time), and how many of them, unfortunately, die.

_{0}represents the final asymptotic value of the cure rate. It is related to the health system’s ability to tackle the infection after adapting to the new outbreak and depends also on other factors like the good health of the citizens. λ

_{1}is related to how fast the adaptation to the emergency was. At the beginning of April, South Korea was already in a post-peak phase of the disease spread. From our initial tests, Equation (13) did not ideally match the data of South Korea, probably because of the more complicated trend, compared to other countries. Therefore, only for the South Korean model, the λ parameter was not constrained by an exponential law. We increased the degrees of freedom of its trend by imposing a sinusoidal law:

_{0}represents the initial value of the mortality rate. The κ

_{0}value is related to the initial health system’s ability to tackle the infection and depends on the good health of the citizens. The κ

_{1}value measures how the rate has changed with time. The mortality rate is supposed to decrease over time, and the higher κ

_{1}is, the faster this decrease.

- S, target time-histories of the susceptible cases,
- E, the target time-histories of the exposed cases,
- I the target time-histories of the infective cases,
- Q, the target time-histories of the quarantined cases,
- R, the target time-histories of the recovered cases,
- D, the target time-histories of the death cases,
- P, the target time-histories of the insusceptible cases.

#### 2.3. Implementation of the SEIR Model with a Stochastic Approach

## 3. Results

## 4. Discussion

_{2}-norm < 0.05). The probable scenarios sometimes presented a wide range of possible solutions because of the intrinsic setting of the stochastic approach. The different scenarios were achieved thanks to a deeper investigation of the model-space domain where the solutions are not driven and influenced by the initial guess of the SEIR model coefficients. One of the main limits of the deterministic approach, instead, is that the results are biased by the selection of the starting point of model parameters.

_{1}) with values around 0.05, followed by Spain (0.044). This confirms the reports which praise the Veneto model, because its administration had the capabilities of testing more quickly than other Italian regions, and the family doctors worked in a stronger synergy with the health structures. It also evidences a lower death rate (κ

_{0}), probably due to the better health system efficiency to treat patients, but also to the greater number of tests. In both data and policies, the Veneto region is more like South Korea than other parts of Italy. These aspects had an impact on the outbreak of the epidemic, as can be seen comparing Figure 4, Figure 5 and Figure 6: The Veneto region is more likely to reach the peak of active cases (Q) before the other regions.

- as we already stated, the problem is underdetermined, so it is preferable to have an acceptable range of values than a unique point value, that could result in being uncertain, as could happen considering only a deterministic approach solution;
- the set of possible predicted scenarios, although related to different solutions with different sets of parameters, are quite similar, thus offering an acceptable level of variability of future predictions.

- We have currently not sufficient information to say that, after recovery, an individual becomes totally immune to the disease, but we made this assumption: the model did not allow the passage from the recovered category to the susceptible category.
- The model does not consider the testing differences between different health system structures and country policies.
- While Italian and Spanish data are well fitted, the South Korean data fitting presents some issues. This evidences that different policies between countries can induce different trends in the spread of the epidemic and that the models should be adapted to different situations, with the introduction or removal of parameters. This would be especially valid in analyzing the situation of the least developed countries, that are not able to afford strict lockdown policies like the developed countries.
- Except for the death rate parameter, the model does not have a strong link to the health resiliency of citizens. The death rate parameter could also be related to external factors like air pollution, which makes people more sensitive to respiratory diseases [19].
- The introduction of Google’s COVID-19 Community Mobility Report represents a constraint that was easily implemented in the model. Further studies on the quality of those data and a rigorous implementation could represent a novel and interesting research topic.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Generalized SEIR (Susceptible-Exposed-Infective-Recovered) model scheme (modified from [5]).

**Figure 2.**Observed data (

**circles**) of quarantined (

**red**), recovered (

**green**), and deaths (

**black**) for Italy, updated on 15 April 2020. The continuous lines refer to the predicted evolution in 30 days according to the SEIR model solved with the deterministic approach.

**Figure 3.**Observed data (

**circles**) of quarantined (

**red**), recovered (

**green**) and deaths (

**black**) for Italy, updated on 15 April 2020. The continuous lines refer to the predicted evolution in 30 days according to the SEIR model solved with the stochastic approach. The solid line refers to the best Particle Swarm Optimization (PSO) solution, the dashed lines refer to the most probable solutions (i.e., the solutions within 5% of the minimum normalized root mean square error (NRMSE)).

**Figure 4.**Observed data (

**circles**) of quarantined (

**red**), recovered (

**green**), and deaths (

**black**) for the Lombardy region, updated on 15 April 2020. The continuous lines refer to the predicted evolution in 30 days according to the SEIR model solved with (

**a**) deterministic approach, (

**b**) stochastic approach. In (

**b**) the solid line refers to the best PSO solution, the dashed lines refer to the most probable solutions (i.e., the solutions within 5% of the minimum NRMSE).

**Figure 5.**Observed data (

**circles**) of quarantined (

**red**), recovered (

**green**), and deaths (

**black**) for Veneto region, updated on 15 April 2020. The continuous lines refer to the predicted evolution in 30 days according to the SEIR model solved with (

**a**) deterministic approach, (

**b**) stochastic approach. In (

**b**) the solid line refers to the best PSO solution, the dashed lines refer to the most probable solutions (i.e., the solutions within 5% of the minimum NRMSE).

**Figure 6.**Observed data (

**circles**) of quarantined (

**red**), recovered (

**green**), and deaths (

**black**) for the Piedmont region, updated on 15 April 2020. The continuous lines refer to the predicted evolution in 30 days according to the SEIR model solved with (

**a**) deterministic approach, (

**b**) stochastic approach. In (

**b**) the solid line refers to the best PSO solution, the dashed lines refer to the most probable solutions (i.e., the solutions within 5% of the minimum NRMSE).

**Figure 7.**Observed data (

**circles**) of quarantined (

**red**), recovered (

**green**), and deaths (

**black**) for Spain, updated on 15 April 2020. The continuous lines refer to the predicted evolution in 30 days according to the SEIR model solved with (

**a**) deterministic approach, (

**b**) stochastic approach. In (

**b**) the solid line refers to the best PSO solution, the dashed lines refer to the most probable solutions (i.e., the solutions within 5% of the minimum NRMSE).

**Figure 8.**Observed data (

**circles**) of quarantined (

**red**), recovered (

**green**), and deaths (

**black**) for South Korea, updated on 15 April 2020. The continuous lines refer to the predicted evolution in 30 days according to the SEIR model solved with (

**a**) deterministic approach, (

**b**) stochastic approach. In (

**b**) the solid line refers to the best PSO solution, the dashed lines refer to the most probable solutions (i.e., the solutions within 5% of the minimum NRMSE).

**Table 1.**Population (approximated) for Italy and Italian regions and for other countries included in the following analysis.

Countries/Regions | Overall Population | Database (Year) |
---|---|---|

Italy | 60,359,546 | Istituto Nazionale di Statistica—ISTAT (2019) |

Lombardy | 10,060,574 | Istituto Nazionale di Statistica—ISTAT (2019) |

Veneto | 4,905,854 | Istituto Nazionale di Statistica—ISTAT (2019) |

Piedmont | 4,356,406 | Istituto Nazionale di Statistica—ISTAT (2019) |

Spain | 47,100,396 | Istituto Nacional de Estadìstìca—INE (2019) |

South Korea | 51,629,512 | Korean Statistical Information Service—KOSIS (November 2018) |

Authors | Country/Region | Date | α | β | γ | δ | λ | κ |
---|---|---|---|---|---|---|---|---|

Peng et al. (2020) [3] | China without Hubei province | 20 Jan–9 Feb | 0.172 | 1 | 0.5 | 0.15 | 0.005–0.04 | 0.005–0.015 |

Peng et al. (2020) | Hubei province without Wuhan city | 20 Jan–9 Feb | 0.133 | 1 | 0.5 | 0.139 | 0.005–0.015 | 0.005–0.02 |

Peng et al. (2020) | Wuhan | 20 Jan–9 Feb | 0.085 | 1 | 0.5 | 0.135 | 0.005–0.015 | 0.005–0.03 |

Peng et al. (2020) | Beijing | 20 Jan–9 Feb | 0.175 | 0.99 | 0.5 | 0.175 | 0.005–0.04 | 0.002 |

Peng et al. (2020) | Shanghai | 20 Jan–9 Feb | 0.183 | 1 | 0.5 | 0.179 | 0.005–0.04 | 0 |

Calafiore et al. (2020) [5] | Italy | 23 Feb–30 Mar | 0.22 | 0.017 | 0.012 | |||

WHO report [10] | China | 12 Feb | 0.1–0.2 | |||||

Dandekar et al. (2020) [11] | Wuhan | 1 Mar–1 Apr | 1 | 0.023 | ||||

Dandekar et al. (2020) | Italy | 1 Mar–1 Apr | 0.74 | 0.032 | ||||

Dandekar et al. (2020) | South Korea | 1 Mar–1 Apr | 0.68 | 0.004 | ||||

Dandekar et al. (2020) | US | 1 Mar–1 Apr | 0.69 | 0.008 | ||||

Shaikh et al. (2020) [12] | India | 14–26 Mar | 0.59 | 0.1 | ||||

Lin et al. (2020) [13] | Wuhan | 15 Jan–24 Feb | 0.59–1.68 | 0.33 | 0.2 | |||

Iwata et al. (2020) [14] | General case | 0.1–1 | 0.07–0.5 | 0.1–1 |

**Table 3.**Coefficients of the PSO best-solutions and deterministic solutions; bold refers to the best solution of the PSO approach (mean and variance on brackets), italic-bold refers to the solution given by the deterministic approach.

Country | α | β * | γ | δ | λ_{0} | λ_{1} | κ_{0} | κ_{1} | NRMSE |
---|---|---|---|---|---|---|---|---|---|

Italy | 0.021(0.086, 0.004) 0.012 | 0.510(1.058, 0.200) 1.170 | 0.265 (0.859,0.226) 1.065 | 0.103 (0.095,0.01) 0.020 | 0.017 (0.017,6.3 × 10 ^{−9})0.017 | 2(1.696, 0.180) 1.983 | 0.029 (0.030,2.7 × 10 ^{−6})0.033 | 0.038 (0.040,3.8 × 10 ^{−6})0.043 | 0.0350.043 |

Spain | 0.037 (0.087,0.002) 0.026 | 1.777 (1.376,0.082) 2 | 0.946(0.954, 0.198) 0.154 | 0.238(0.095, 0.004) 0.614 | 0.044(0.044, 6.8 × 10 ^{−7})0.043 | 0.156(0.159, 0.0004) 0.160 | 0.030(0.030, 1.6 × 10 ^{−6})0.028 | 0.046(0.047, 3.7 × 10 ^{−6})0.044 | 0.0460.052 |

South Korea | 0.292 (0.270 0.0004)0.1 | 2(1.915, 0.009) 0.974 | 2(1.846, 0.046) 1.902 | 0.123(0.136, 7.8 × 10 ^{−5})0.313 | 0.05 ** | - | 8.3 × 10^{−4}(8.3 × 10 ^{−4}, 3.5 × 10^{−11})0.007 | 7 × 10^{−6}(2.1 × 10 ^{−6}, 2.1 × 10^{−11})0.134 | 0.0740.078 |

Lombardy | 0(0.132, 0.012) 8.9 × 10^{−4} | 0.460(1.658, 0.188) 0.81 | 0.295(1.093, 0.531) 0.302 | 0.145(0.198, 0.065) 0.253 | 0.027(0.026, 3 × 10 ^{−8})0.027 | 0.981(1.576, 0.247) 1.925 | 0.036(0.036, 6.9 × 10 ^{−6})0.045 | 0.031(0.031, 6.9 × 10 ^{−6})0.0405 | 0.0620.061 |

Veneto | 0.133(0.102, 0.002) 0.049 | 1.704(1.175, 0.190) 0.97 | 0.920(0.698, 0.144) 0.246 | 0.032(0.034, 0.0004) 0.09 | 0.049(0.182, 0.093) 0.099 | 0.009(0.008, 0.000) 0.004 | 0.008(0.008, 6.4 × 10 ^{−8})0.009 | 0.0215(0.021, 1.3 × 10 ^{−6})0.024 | 0.0350.040 |

Piedmont | 0.240(0.163, 0.009) 0 | 1.990(1.518, 0.232) 0.994 | 0.265(1.191, 0.374) 0.195 | 0.012(0.117, 0.052) 0.344 | 0.386(0.309, 0.104) 0.069 | 0.001(0.005, 2.5 × 10 ^{−5})0.007 | 0.019(0.018, 3.7 × 10 ^{−6})0.019 | 0.034(0.031, 1.9 × 10 ^{−5})0.035 | 0.0560.050 |

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**MDPI and ACS Style**

Godio, A.; Pace, F.; Vergnano, A.
SEIR Modeling of the Italian Epidemic of SARS-CoV-2 Using Computational Swarm Intelligence. *Int. J. Environ. Res. Public Health* **2020**, *17*, 3535.
https://doi.org/10.3390/ijerph17103535

**AMA Style**

Godio A, Pace F, Vergnano A.
SEIR Modeling of the Italian Epidemic of SARS-CoV-2 Using Computational Swarm Intelligence. *International Journal of Environmental Research and Public Health*. 2020; 17(10):3535.
https://doi.org/10.3390/ijerph17103535

**Chicago/Turabian Style**

Godio, Alberto, Francesca Pace, and Andrea Vergnano.
2020. "SEIR Modeling of the Italian Epidemic of SARS-CoV-2 Using Computational Swarm Intelligence" *International Journal of Environmental Research and Public Health* 17, no. 10: 3535.
https://doi.org/10.3390/ijerph17103535