{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T08:27:33Z","timestamp":1762072053164,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,10,13]],"date-time":"2022-10-13T00:00:00Z","timestamp":1665619200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"The Center for Research and Development in Mathematics and Applications of University of Aveiro","award":["CEECIND\/00564\/2018"],"award-info":[{"award-number":["CEECIND\/00564\/2018"]}]},{"name":"Portuguese Foundation for Science and Technology (FCT\u2013Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia)","award":["CEECIND\/00564\/2018"],"award-info":[{"award-number":["CEECIND\/00564\/2018"]}]},{"name":"Thematic Line BioMath from CIDMA","award":["CEECIND\/00564\/2018"],"award-info":[{"award-number":["CEECIND\/00564\/2018"]}]},{"name":"FCT Researcher Program CEEC Individual 2018","award":["CEECIND\/00564\/2018"],"award-info":[{"award-number":["CEECIND\/00564\/2018"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["MCA"],"abstract":"<jats:p>The COVID-19 pandemic remains a global problem that affects the health of millions of people and the world economy. Identifying how the movement of people between regions of the world, countries, and municipalities and how the close contact between individuals of different age groups promotes the spread of infectious diseases is a pressing concern for society, during epidemic outbreaks and pandemics, such as COVID-19. Networks and Graph Theory provide adequate and powerful tools to study the spread of communicable diseases. In this work, we use Graph Theory to analyze COVID-19 transmission dynamics between municipalities of Aveiro district, in Portugal, and between different age groups, considering data from 2020 and 2021, in order to better understand the spread of this disease, as well as preparing actions for possible future pandemics. We used a digraph structure that models the transmission of SARS-CoV-2 virus between Aveiro\u2019s municipalities and between age groups. To understand how a node fits over the contact digraphs, we studied centrality measures, namely eigencentrality, closeness, degree, and betweenness. Transmission ratios were also considered to determine whether there were certain age groups or municipals that were more responsible for the virus\u2019s spread. According to the results of this research, transmissions mostly occur within the same social groupings, that is, within the same municipalities and age groups. However, the study of centrality measures, eliminating loops, reveals that municipalities such as Aveiro, Estarreja and Ovar are relevant nodes in the transmission network of municipalities as well as the age group of 40\u201349 in the transmission network of age groups. Furthermore, we conclude that vaccination is effective in reducing the virus.<\/jats:p>","DOI":"10.3390\/mca27050086","type":"journal-article","created":{"date-parts":[[2022,10,13]],"date-time":"2022-10-13T22:21:11Z","timestamp":1665699671000},"page":"86","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Graph Theory Approach to COVID-19 Transmission by Municipalities and Age Groups"],"prefix":"10.3390","volume":"27","author":[{"given":"Pedro","family":"Machado","sequence":"first","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"given":"Sofia J.","family":"Pinheiro","sequence":"additional","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1051-8084","authenticated-orcid":false,"given":"Vera","family":"Afreixo","sequence":"additional","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7238-546X","authenticated-orcid":false,"given":"Cristiana J.","family":"Silva","sequence":"additional","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1009-6611","authenticated-orcid":false,"given":"Rui","family":"Leit\u00e3o","sequence":"additional","affiliation":[{"name":"Public Health Unit, Baixo Vouga Health Centers Cluster, 3810-164 Aveiro, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"e230","DOI":"10.1017\/S095026882000223X","article-title":"Social Network Analysis of COVID-19 Transmission in Karnataka, India","volume":"148","author":"Saraswathi","year":"2020","journal-title":"Epidemiol. Infect."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1007\/s15010-020-01516-2","article-title":"COVID-19 outbreak: History, mechanism, transmission, structural studies and therapeutics","volume":"49","author":"Yesudhas","year":"2021","journal-title":"Infection"},{"key":"ref_3","unstructured":"Li, C., and Tang, H. (2022, July 06). Since January 2020 Elsevier Has Created a COVID-19 Resource Centre with Free Information in English and Mandarin on the Novel Coronavirus COVID-19. The COVID-19 Resource Centre Is Hosted on Elsevier Connect, the Company\u2019s Public News and Information Website. Available online: https:\/\/cdn.who.int\/media\/docs\/default-source\/whhd-2021\/scientific-publications\/2.jhi_5may2021.pdf?sfvrsn=6526a2a5_5."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"8581","DOI":"10.1038\/s41598-021-87837-0","article-title":"A social network analysis of the spread of COVID-19 in South Korea and policy implications","volume":"11","author":"Jo","year":"2021","journal-title":"Sci. Rep."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1103\/RevModPhys.87.925","article-title":"Epidemic processes in complex networks","volume":"87","author":"Castellano","year":"2015","journal-title":"Rev. Mod. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2016.10.006","article-title":"Statistical physics of vaccination","volume":"664","author":"Wang","year":"2016","journal-title":"Phys. Rep."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2016.05.004","article-title":"Vital nodes identification in complex networks","volume":"650","author":"Chen","year":"2016","journal-title":"Phys. Rep."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1209\/epl\/i2004-10286-2","article-title":"Efficient local strategies for vaccination and network attack","volume":"68","author":"Holme","year":"2004","journal-title":"Europhys. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1209\/0295-5075\/99\/68007","article-title":"Identifying influential spreaders and efficiently estimating infection numbers in epidemic models: A walk counting approach","volume":"99","author":"Bauer","year":"2012","journal-title":"EPL"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"032812","DOI":"10.1103\/PhysRevE.90.032812","article-title":"Role of centrality for the identification of influential spreaders in complex networks","volume":"90","author":"Barbieri","year":"2014","journal-title":"Phys. Rev. E Stat. Nonlinear Soft Matter Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3544","DOI":"10.1098\/rspb.2011.0290","article-title":"Using network properties to predict disease dynamics on human contact networks","volume":"278","author":"Ames","year":"2011","journal-title":"Proc. R. Soc. B Biol. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Freitas, A., Escudeiro, S., Reis, J., and Silva, C.J. (2022). COVID-19\u2019s pandemic: A new way of thinking through linear combinations of proportions. Mathematical Analysis of Infectious Diseases, Academic Press.","DOI":"10.1016\/B978-0-32-390504-6.00008-5"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1021\/ci00047a033","article-title":"Applications of Graph Theory in Chemistry","volume":"25","author":"Balaban","year":"1985","journal-title":"J. Chem. Inf. Comput. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"103827","DOI":"10.1016\/j.scs.2022.103827","article-title":"Generation of optimal (de)centralized layouts for urban drainage systems: A graph-theory-based combinatorial multi-objective optimization framework","volume":"81","author":"Hesarkazzazi","year":"2021","journal-title":"Sustain. Cities Soc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"585","DOI":"10.3389\/fnins.2019.00585","article-title":"Application of graph theory for identifying connectivity patterns in human brain networks: A systematic review","volume":"13","author":"Farahani","year":"2019","journal-title":"Front. Neurosci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"bbac230","DOI":"10.1093\/bib\/bbac230","article-title":"Disease spreading modeling and analysis: A survey","volume":"23","author":"Petrizzelli","year":"2022","journal-title":"Brief. Bioinform."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1007\/s40812-020-00164-9","article-title":"The Italian value chain in the pandemic: The input\u2013output impact of Covid-19 lockdown","volume":"47","author":"Giammetti","year":"2020","journal-title":"J. Ind. Bus. Econ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2025008","DOI":"10.1080\/21645515.2021.2025008","article-title":"The dramatic increase in anti-vaccine discourses during the COVID-19 pandemic: A social network analysis of Twitter","volume":"18","author":"Durmaz","year":"2022","journal-title":"Hum. Vaccines Immunother."},{"key":"ref_19","first-page":"1","article-title":"A study of plithogenic graphs: Applications in spreading coronavirus disease (COVID-19) globally","volume":"6","author":"Sultana","year":"2022","journal-title":"J. Ambient Intell. Humaniz. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Aletti, G., Benfenati, A., and Naldi, G. (2021). Graph, spectra, control and epidemics: An example with a SEIR model. Mathematics, 9.","DOI":"10.3390\/math9222987"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1109\/MCS.2015.2495000","article-title":"Analysis and Control of Epidemics: A Survey of Spreading Processes on Complex Networks","volume":"36","author":"Nowzari","year":"2016","journal-title":"IEEE Control Syst. Mag."},{"key":"ref_22","unstructured":"Cardoso, D.M., Szymanski, J., and Rostami, M. (2009). Matem\u00e1tica Discreta: Combinat\u00f3ria, Teoria dos Grafos e Algoritmos, Escolar Editora."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Hansen, D.L., Shneiderman, B., Smith, M.A., and Himelboim, I. (2020). Calculating and visualizing network metrics. Analyzing Social Media Networks with NodeXL, Elsevier.","DOI":"10.1016\/B978-0-12-817756-3.00006-6"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Golbeck, J. (2015). Analyzing networks. Introduction to Social Media Investigation, Elsevier.","DOI":"10.1016\/B978-0-12-801656-5.00021-4"},{"key":"ref_25","unstructured":"Csardi, G., and Nepusz, T. (2022, July 05). The Igraph Software Package for Complex Network Research. Available online: https:\/\/igraph.org."}],"container-title":["Mathematical and Computational Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2297-8747\/27\/5\/86\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:53:26Z","timestamp":1760144006000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2297-8747\/27\/5\/86"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,13]]},"references-count":25,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["mca27050086"],"URL":"https:\/\/doi.org\/10.3390\/mca27050086","relation":{},"ISSN":["2297-8747"],"issn-type":[{"type":"electronic","value":"2297-8747"}],"subject":[],"published":{"date-parts":[[2022,10,13]]}}}