# Complex Network Modelling of Origin–Destination Commuting Flows for the COVID-19 Epidemic Spread Analysis in Italian Lombardy Region

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Origin–Destination Networks

#### 2.2. Epidemic Spread on Networks

- transmission from an infected node to a susceptible node occurs across an edge as a Poisson process with rate $\beta $;
- an infected node recovers by following a Poisson process with rate $\gamma $.

#### 2.3. Statistical Analysis

- value of the peak;
- time at which the peak occurs;
- area under the curve.

## 3. Results

#### 3.1. Network Structure

#### 3.2. Simulations of Epidemic Spread

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Paules, C.I.; Marston, H.D.; Fauci, A.S. Coronavirus infections—More than just the common cold. JAMA
**2020**, 323, 707–708. [Google Scholar] [CrossRef] [Green Version] - World Health Organization. Coronavirus Disease 2019 (COVID-19): Situation Report, 82; World Health Organization: Geneva, Switzerland, 2020. [Google Scholar]
- World Health Organization. WHO Director-General’s Opening Remarks at the Media Briefing on COVID19- March 2020; World Health Organization: Geneva, Switzerland, 2020. [Google Scholar]
- Available online: https://coronavirus.jhu.edu/map.html (accessed on 7 April 2021).
- Report45, Monitoraggio Fase2 (DM Salute 20 Aprile 2020), Dati Relativi alla Settimana 15/3/2021–21/3/2021; Ministero della Salute, Istituto Superiore di Sanitá, Cabina di Regia ai sensi del DM Salute: Rome, Italy, 2020.
- Pastor-Satorras, R.; Castellano, C.; Van Mieghem, P.; Vespignani, A. Epidemic processes in complex networks. Rev. Mod. Phys.
**2015**, 87, 925. [Google Scholar] [CrossRef] [Green Version] - Colizza, V.; Barrat, A.; Barthélemy, M.; Vespignani, A. The role of the airline transportation network in the prediction and predictability of global epidemics. Proc. Natl. Acad. Sci. USA
**2006**, 103, 2015–2020. [Google Scholar] [CrossRef] [Green Version] - Colizza, V.; Barrat, A.; Barthélemy, M.; Vespignani, A. The modeling of global epidemics: Stochastic dynamics and predictability. Bull. Math. Biol.
**2006**, 68, 1893–1921. [Google Scholar] [CrossRef] - Ni, S.; Weng, W. Impact of travel patterns on epidemic dynamics in heterogeneous spatial metapopulation networks. Phys. Rev. E
**2009**, 79, 016111. [Google Scholar] [CrossRef] - Nouvellet, P.; Bhatia, S.; Cori, A.; Ainslie, K.E.; Baguelin, M.; Bhatt, S.; Boonyasiri, A.; Brazeau, N.F.; Cattarino, L.; Cooper, L.V.; et al. Reduction in mobility and COVID-19 transmission. Nat. Commun.
**2021**, 12, 1–9. [Google Scholar] [CrossRef] [PubMed] - Colizza, V.; Barrat, A.; Barthelemy, M.; Valleron, A.J.; Vespignani, A. Modeling the worldwide spread of pandemic influenza: Baseline case and containment interventions. PLoS Med.
**2007**, 4, e13. [Google Scholar] [CrossRef] [Green Version] - Bowen, J.T., Jr.; Laroe, C. Airline networks and the international diffusion of severe acute respiratory syndrome (SARS). Geogr. J.
**2006**, 172, 130–144. [Google Scholar] [CrossRef] - Chinazzi, M.; Davis, J.T.; Ajelli, M.; Gioannini, C.; Litvinova, M.; Merler, S.; Y Piontti, A.P.; Mu, K.; Rossi, L.; Sun, K.; et al. The effect of travel restrictions on the spread of the 2019 novel coronavirus (COVID-19) outbreak. Science
**2020**, 368, 395–400. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Amini, B.; Peiravian, F.; Mojarradi, M.; Derrible, S. Comparative analysis of traffic performance of urban transportation systems. Transp. Res. Rec.
**2016**, 2594, 159–168. [Google Scholar] [CrossRef] - Tak, S.; Kim, S.; Byon, Y.J.; Lee, D.; Yeo, H. Measuring health of highway network configuration against dynamic Origin–Destination demand network using weighted complex network analysis. PLoS ONE
**2018**, 13, e0206538. [Google Scholar] [CrossRef] [PubMed] - Newman, M.E. The structure and function of complex networks. SIAM Rev.
**2003**, 45, 167–256. [Google Scholar] [CrossRef] [Green Version] - Boccaletti, S.; Latora, V.; Moreno, Y.; Chavez, M.; Hwang, D.U. Complex networks: Structure and dynamics. Phys. Rep.
**2006**, 424, 175–308. [Google Scholar] [CrossRef] - Barrat, A.; Barthelemy, M.; Pastor-Satorras, R.; Vespignani, A. The architecture of complex weighted networks. Proc. Natl. Acad. Sci. USA
**2004**, 101, 3747–3752. [Google Scholar] [CrossRef] [Green Version] - Anderson, R.M.; Anderson, B.; May, R.M. Infectious Diseases of Humans: Dynamics and Control; Oxford University Press: Oxford, UK, 1992. [Google Scholar]
- Koziol, K.; Stanislawski, R.; Bialic, G. Fractional-Order SIR Epidemic Model for Transmission Prediction of COVID-19 Disease. Appl. Sci.
**2020**, 10, 8316. [Google Scholar] [CrossRef] - Kiss, I.Z.; Miller, J.C.; Simon, P.L. Mathematics of Epidemics on Networks; Springer: Cham, Switzerland, 2017; Volume 598. [Google Scholar]
- Atkeson, A. What Will Be the Economic Impact of COVID-19 in the US? Rough Estimates of Disease Scenarios; Technical Report; National Bureau of Economic Research: Cambridge, MA USA, 2020. [Google Scholar]
- Remuzzi, A.; Remuzzi, G. COVID-19 and Italy: What next? Lancet
**2020**, 395, 1225–1228. [Google Scholar] [CrossRef] - Anderson, R.M.; Heesterbeek, H.; Klinkenberg, D.; Hollingsworth, T.D. How will country-based mitigation measures influence the course of the COVID-19 epidemic? Lancet
**2020**, 395, 931–934. [Google Scholar] [CrossRef] - Guimera, R.; Mossa, S.; Turtschi, A.; Amaral, L.N. The worldwide air transportation network: Anomalous centrality, community structure, and cities’ global roles. Proc. Natl. Acad. Sci. USA
**2005**, 102, 7794–7799. [Google Scholar] [CrossRef] [Green Version] - Barabási, A.L. Scale-free networks: A decade and beyond. Science
**2009**, 325, 412–413. [Google Scholar] [CrossRef] [Green Version] - La Gatta, V.; Moscato, V.; Postiglione, M.; Sperli, G. An Epidemiological Neural network exploiting Dynamic Graph Structured Data applied to the COVID-19 outbreak. IEEE Trans. Big Data
**2020**. [Google Scholar] [CrossRef] - Hâncean, M.G.; Slavinec, M.; Perc, M. The impact of human mobility networks on the global spread of COVID-19. J. Complex Netw.
**2020**, 8, cnaa041. [Google Scholar] [CrossRef] - Dehning, J.; Zierenberg, J.; Spitzner, F.P.; Wibral, M.; Neto, J.P.; Wilczek, M.; Priesemann, V. Inferring change points in the spread of COVID-19 reveals the effectiveness of interventions. Science
**2020**, 369. [Google Scholar] [CrossRef] - Zhong, L.; Mu, L.; Li, J.; Wang, J.; Yin, Z.; Liu, D. Early prediction of the 2019 novel coronavirus outbreak in the mainland China based on simple mathematical model. IEEE Access
**2020**, 8, 51761–51769. [Google Scholar] [CrossRef] [PubMed] - Liu, M.; Thomadsen, R.; Yao, S. Forecasting the spread of COVID-19 under different reopening strategies. Sci. Rep.
**2020**, 10, 1–8. [Google Scholar] [CrossRef] - Gatto, M.; Bertuzzo, E.; Mari, L.; Miccoli, S.; Carraro, L.; Casagrandi, R.; Rinaldo, A. Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures. Proc. Natl. Acad. Sci. USA
**2020**, 117, 10484–10491. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**Degree distributions (reported in logarithm scale in the inset) for the three networks. The degree k corresponds to the number of urban areas connected to each zone in the OD matrices. The distributions are heavy-tailed.

**Figure 2.**Strength distributions (reported in logarithm scale in the inset) for the three networks. The strength s corresponds to the sum of the flows among the connected urban areas. The distributions are heavy-tailed.

**Figure 4.**SIR curves for (

**A**) study network, (

**B**) work network, (

**C**) occasional network, (

**D**) total network computed for ${R}_{0}=2$.

**Figure 5.**Violin plots for the distributions of (

**A**) the peak values of $I\left(t\right)$, (

**B**) the temporal locations of the peaks of $I\left(t\right)$, (

**C**) the area under $I\left(t\right)$ computed for ${R}_{0}=2$.

**Figure 6.**Error bars (mean value and standard deviation) for the distributions of (

**A**)the peak values of $I\left(t\right)$, (

**B**) the temporal location of the peak of $I\left(t\right)$, (

**C**) the area under $I\left(t\right)$ computed for different values of ${R}_{0}$.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lombardi, A.; Amoroso, N.; Monaco, A.; Tangaro, S.; Bellotti, R.
Complex Network Modelling of Origin–Destination Commuting Flows for the COVID-19 Epidemic Spread Analysis in Italian Lombardy Region. *Appl. Sci.* **2021**, *11*, 4381.
https://doi.org/10.3390/app11104381

**AMA Style**

Lombardi A, Amoroso N, Monaco A, Tangaro S, Bellotti R.
Complex Network Modelling of Origin–Destination Commuting Flows for the COVID-19 Epidemic Spread Analysis in Italian Lombardy Region. *Applied Sciences*. 2021; 11(10):4381.
https://doi.org/10.3390/app11104381

**Chicago/Turabian Style**

Lombardi, Angela, Nicola Amoroso, Alfonso Monaco, Sabina Tangaro, and Roberto Bellotti.
2021. "Complex Network Modelling of Origin–Destination Commuting Flows for the COVID-19 Epidemic Spread Analysis in Italian Lombardy Region" *Applied Sciences* 11, no. 10: 4381.
https://doi.org/10.3390/app11104381