# The Critical Role of Networks to Describe Disease Spreading Dynamics in Social Systems: A Perspective

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## Abstract

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## 1. Introduction

#### 1.1. Epidemic Spreading in Scale-Free Networks

#### 1.2. Spreading, Node Clustering Coefficient, and Node Assortativity

#### 1.3. Spreading and Community Structure

#### 1.4. Effective Network Size (ENS)

#### 1.5. The Case of the COVID-19 Spreading

#### 1.6. Predict Epidemic Spreading in Real-World Social Networks

## 2. Discussion

## 3. Conclusions

## Funding

## Conflicts of Interest

## References

- Pastor-Satorras, R.; Castellano, C.; Van Mieghem, P.; Vespignani, A. Epidemic Processes in Complex Networks. Rev. Mod. Phys.
**2015**, 87, 925. [Google Scholar] [CrossRef] - Thurner, S.; Klimek, P.; Hanel, R. A Network-Based Explanation of Why Most COVID-19 Infection Curves Are Linear. Proc. Natl. Acad. Sci. USA
**2020**, 117, 22684–22689. [Google Scholar] [CrossRef] - Keeling, M.J.; Eames, K.T.D. Networks and Epidemic Models. J. R. Soc. Interface
**2005**, 2, 295–307. [Google Scholar] [CrossRef] [PubMed] - Manzo, G. Complex Social Networks Are Missing in the Dominant COVID-19 Epidemic Models. Sociologica
**2020**, 14, 31–49. [Google Scholar] [CrossRef] - Salathé, M.; Jones, J.H. Dynamics and Control of Diseases in Networks with Community Structure. PLoS Comput. Biol.
**2010**, 6, e1000736. [Google Scholar] [CrossRef] [PubMed] - Ferguson, N.M.; Laydon, D.; Nedjati-Gilani, G.; Imai, N.; Ainslie, K.; Baguelin, M.; Bhatia, S.; Boonyasiri, A.; Cucunubá, Z.; Cuomo-Dannenburg, G.; et al. Report 9: Impact of Non-Pharmaceutical Interventions (NPIs) to Reduce COVID19 Mortality and Healthcare Demand. Imp. Coll. Lond.
**2020**, 10, 491–497. [Google Scholar] [CrossRef] - Kissler, S.; Tedijanto, C.; Lipsitch, M.; Grad, Y. Social Distancing Strategies for Curbing the COVID-19 Epidemic. MedRxiv
**2020**. [Google Scholar] [CrossRef] - Nishi, A.; Dewey, G.; Endo, A.; Neman, S.; Iwamoto, S.K.; Ni, M.Y.; Tsugawa, Y.; Iosifidis, G.; Smith, J.D.; Young, S.D. Network Interventions for Managing the COVID-19 Pandemic and Sustaining Economy. Proc. Natl. Acad. Sci. USA
**2020**, 117, 30285–30294. [Google Scholar] [CrossRef] - Bellingeri, M.; Turchetto, M.; Bevacqua, D.; Scotognella, F.; Alfieri, R.; Nguyen, Q.; Cassi, D. Modeling the Consequences of Social Distancing Over Epidemics Spreading in Complex Social Networks: From Link Removal Analysis to SARS-CoV-2 Prevention. Front. Phys.
**2021**, 9, 681343. [Google Scholar] [CrossRef] - Firth, J.A.; Hellewell, J.; Klepac, P.; Kissler, S.; Jit, M.; Atkins, K.E.; Clifford, S.; Villabona-Arenas, C.J.; Meakin, S.R.; Diamond, C.; et al. Using a Real-World Network to Model Localized COVID-19 Control Strategies. Nat. Med.
**2020**, 26, 1616–1622. [Google Scholar] [CrossRef] - Chung, N.N.; Chew, L.Y. Modelling Singapore COVID-19 Pandemic with a SEIR Multiplex Network Model. Sci. Rep.
**2021**, 11, 10122. [Google Scholar] [CrossRef] [PubMed] - Pastor-Satorras, R.; Vespignani, A. Epidemic Spreading in Scale-Free Networks. Phys. Rev. Lett.
**2001**, 86, 3200–3203. [Google Scholar] [CrossRef] [PubMed] - Albert, R.; Barabási, A.L. Statistical Mechanics of Complex Networks. Rev. Mod. Phys.
**2002**, 74, 47. [Google Scholar] [CrossRef] - McCabe, C.M.; Nunn, C.L. Effective Network Size Predicted from Simulations of Pathogen Outbreaks through Social Networks Provides a Novel Measure of Structure-Standardized Group Size. Front. Vet. Sci.
**2018**, 5, 71. [Google Scholar] [CrossRef] [PubMed] - Borgatti, S.P.; Mehra, A.; Brass, D.J.; Labianca, G. Network Analysis in the Social Sciences. Science
**2009**, 323, 892–895. [Google Scholar] [CrossRef] [PubMed] - Bellingeri, M.; Bevacqua, D.; Sartori, F.; Turchetto, M.; Scotognella, F.; Alfieri, R.; Nguyen, N.K.K.; Le, T.T.; Nguyen, Q.; Cassi, D. Considering Weights in Real Social Networks: A Review. Front. Phys.
**2023**, 11, 242. [Google Scholar] [CrossRef] - Opsahl, T.; Panzarasa, P. Clustering in Weighted Networks. Soc. Netw.
**2009**, 31, 155–163. [Google Scholar] [CrossRef] - Holland, P.W.; Leinhardt, S. Transitivity in Structural Models of Small Groups. Comp. Group. Stud.
**1971**, 2, 107–124. [Google Scholar] [CrossRef] - Watts, D.J.; Strogatz, S.H. Collective Dynamics of ‘Small-World’ Networks. Nature
**1998**, 393, 440–442. [Google Scholar] [CrossRef] - Noldus, R.; Mieghem, P. Van Assortativity in Complex Networks. J. Complex Netw.
**2014**, 3, 507–542. [Google Scholar] [CrossRef] - Newman, M.E.J. Mixing Patterns in Networks. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top.
**2003**, 67, 026126. [Google Scholar] [CrossRef] [PubMed] - Fisher, D.; Silk, M.; Franks, D. The Perceived Assortativity of Social Networks: Methodological Problems and Solutions. In Trends in Social Network Analysis; Springer: Berlin/Heidelberg, Germany, 2017; pp. 1–19. [Google Scholar]
- Badham, J.; Stocker, R. The Impact of Network Clustering and Assortativity on Epidemic Behaviour. Theor. Popul. Biol.
**2010**, 77, 71–75. [Google Scholar] [CrossRef] [PubMed] - Rapoport, A.; Horvath, W.J. A Study of a Large Sociogram. Behav. Sci.
**2007**, 6, 279–291. [Google Scholar] [CrossRef] [PubMed] - Volz, E.M.; Miller, J.C.; Galvani, A.; Meyers, L. Effects of Heterogeneous and Clustered Contact Patterns on Infectious Disease Dynamics. PLoS Comput. Biol.
**2011**, 7, e1002042. [Google Scholar] [CrossRef] - Fransson, C.; Trapman, P. SIR Epidemics and Vaccination on Random Graphs with Clustering. J. Math. Biol.
**2019**, 78, 2369–2398. [Google Scholar] [CrossRef] [PubMed] - Kumpula, J.M.; Onnela, J.P.; Saramäki, J.; Kertész, J.; Kaski, K. Model of Community Emergence in Weighted Social Networks. Comput. Phys. Commun.
**2009**, 180, 517–522. [Google Scholar] [CrossRef] - Clauset, A.; Newman, M.E.J.; Moore, C. Finding Community Structure in Very Large Networks. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top.
**2004**, 70, 066111. [Google Scholar] [CrossRef] [PubMed] - Sartori, F.; Turchetto, M.; Bellingeri, M.; Scotognella, F.; Alfieri, R.; Nguyen, N.K.K.; Le, T.T.; Nguyen, Q.; Cassi, D. A Comparison of Node Vaccination Strategies to Halt SIR Epidemic Spreading in Real-World Complex Networks. Sci. Rep.
**2022**, 12, 21355. [Google Scholar] [CrossRef] - Wang, Z.; Zhao, D.W.; Wang, L.; Sun, G.Q.; Jin, Z. Immunity of Multiplex Networks via Acquaintance Vaccination. Europhys. Lett.
**2015**, 112, 48002. [Google Scholar] [CrossRef] - Holme, P. Efficient Local Strategies for Vaccination and Network Attack. Europhys. Lett.
**2004**, 68, 908. [Google Scholar] [CrossRef] - Bellingeri, M.; Bevacqua, D.; Turchetto, M.; Scotognella, F.; Alfieri, R.; Nguyen, N.K.K.; Le, T.T.; Nguyen, Q.; Cassi, D. Network Structure Indexes to Forecast Epidemic Spreading in Real-World Complex Networks. Front. Phys.
**2022**, 10, 1121. [Google Scholar] [CrossRef] - Freeman, L.C. A Set of Measures of Centrality Based on Betweenness. Sociometry
**1977**, 40, 35–41. [Google Scholar] [CrossRef] - Bonchev, D.; Buck, G.A. Quantitative Measures of Network Complexity. In Complexity in Chemistry, Biology, and Ecology; Bonchev, D., Rouvray, D.H., Eds.; Springer: Boston, MA, USA, 2005; pp. 191–235. ISBN 978-0-387-25871-3. [Google Scholar]
- Génois, M.; Barrat, A. Can Co-Location Be Used as a Proxy for Face-to-Face Contacts? EPJ Data Sci.
**2018**, 7, 11. [Google Scholar] [CrossRef] - Ozella, L.; Paolotti, D.; Lichand, G.; Rodríguez, J.P.; Haenni, S.; Phuka, J.; Leal-Neto, O.B.; Cattuto, C. Using Wearable Proximity Sensors to Characterize Social Contact Patterns in a Village of Rural Malawi. EPJ Data Sci.
**2021**, 10, 46. [Google Scholar] [CrossRef] - Vanhems, P.; Barrat, A.; Cattuto, C.; Pinton, J.F.; Khanafer, N.; Régis, C.; Kim, B.A.; Comte, B.; Voirin, N. Estimating Potential Infection Transmission Routes in Hospital Wards Using Wearable Proximity Sensors. PLoS ONE
**2013**, 8, e73970. [Google Scholar] [CrossRef] - Klise, K.; Beyeler, W.; Finley, P.; Makvandi, M. Analysis of Mobility Data to Build Contact Networks for COVID-19. PLoS ONE
**2021**, 16, e0249726. [Google Scholar] [CrossRef] [PubMed] - Oliver, N.; Lepri, B.; Sterly, H.; Lambiotte, R.; Deletaille, S.; De Nadai, M.; Letouzé, E.; Salah, A.A.; Benjamins, R.; Cattuto, C.; et al. Mobile Phone Data for Informing Public Health Actions across the COVID-19 Pandemic Life Cycle. Sci. Adv.
**2020**, 6, eabc0764. [Google Scholar] [CrossRef] - Ciddio, M.; Mari, L.; Sokolow, S.H.; De Leo, G.A.; Casagrandi, R.; Gatto, M. The Spatial Spread of Schistosomiasis: A Multidimensional Network Model Applied to Saint-Louis Region, Senegal. Adv. Water Resour.
**2017**, 108, 406–415. [Google Scholar] [CrossRef]

**Figure 1.**

**Node clustering**. (

**A**) Toy model network with a lower clustering coefficient (one closed triplet) vs. (

**B**) network with a higher clustering coefficient (four closed triplets). Links of closed triplets are in red.

**Node assortativity**. (

**C**) A disassortative network in which nodes of higher degree (more links) are connected preferentially with lower-degree nodes. (

**D**) Assortative network in which nodes of higher degree are connected preferentially with nodes of higher degree and nodes of lower degree are connected preferentially with lower-degree nodes.

**Network community structure**. (

**E**) Random network that does not present a community structure. (

**F**) Network with a strong community structure (node color indicates nodes belonging to the same community); this network comprises four clearly separated communities.

**Figure 2.**

**The effective network size**. (

**A**) A hypothetical real network over which a disease can spread; (

**B**) the complete network underlying the mean-field approach at the base of the classic CMs, in which every node interacts with each other. The authors simulated the spread of epidemics via an SIR-type model with the same infectious probability over the real (

**A**) and the complete network of the same size (

**B**). The complete network shows a higher pace of the disease spreading, resulting in a shorter outbreak duration. Starting from the real network in (

**A**), to obtain a similar SIR spreading pace measured by the outbreak duration, we must consider a complete network of larger size, such as the one depicted in (

**C**). On the other hand, to produce the same spreading pace in the real (

**A**) and the corresponding mean-field approach complete network (

**B**), we must assume different infection probabilities over the network links, decreasing the infection probability in the complete network. Assuming the complete network in (

**C**) as the complete network corresponding to the outbreak characteristics of the real network in (

**A**), its number of nodes is the “effective network size” (ENS) of the real network in (

**A**) [14].

**Node distance and the pace of the epidemic spreading**. (

**D**) The chain network of $N$ = 9 nodes; (

**E**) the star network of $N$ = 9 nodes. The two model networks have the same number of links $L$, i.e., $L=N-1=8$, and, for this, the same average node degree $\overline{k}$. The two network models are limit structures showing very different distances among nodes. The chain network is much longer than the star network (average node distance $\overline{d}$ = 1.8 for the star and $\overline{d}$ = 3.3 for the chain network). Consequently, the chain network will show a lower epidemic spreading pace.

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**MDPI and ACS Style**

Bellingeri, M.; Bevacqua, D.; Scotognella, F.; Cassi, D.
The Critical Role of Networks to Describe Disease Spreading Dynamics in Social Systems: A Perspective. *Mathematics* **2024**, *12*, 792.
https://doi.org/10.3390/math12060792

**AMA Style**

Bellingeri M, Bevacqua D, Scotognella F, Cassi D.
The Critical Role of Networks to Describe Disease Spreading Dynamics in Social Systems: A Perspective. *Mathematics*. 2024; 12(6):792.
https://doi.org/10.3390/math12060792

**Chicago/Turabian Style**

Bellingeri, Michele, Daniele Bevacqua, Francesco Scotognella, and Davide Cassi.
2024. "The Critical Role of Networks to Describe Disease Spreading Dynamics in Social Systems: A Perspective" *Mathematics* 12, no. 6: 792.
https://doi.org/10.3390/math12060792