{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:34:16Z","timestamp":1760146456991,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,11,5]],"date-time":"2024-11-05T00:00:00Z","timestamp":1730764800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"region of Normandy","award":["Mod\u00e9lisation et Analyse des Syst\u00e8mes Complexes en Biologie (MASyComB)","PAPIIT IN107624"],"award-info":[{"award-number":["Mod\u00e9lisation et Analyse des Syst\u00e8mes Complexes en Biologie (MASyComB)","PAPIIT IN107624"]}]},{"DOI":"10.13039\/501100005739","name":"UNAM","doi-asserted-by":"publisher","award":["Mod\u00e9lisation et Analyse des Syst\u00e8mes Complexes en Biologie (MASyComB)","PAPIIT IN107624"],"award-info":[{"award-number":["Mod\u00e9lisation et Analyse des Syst\u00e8mes Complexes en Biologie (MASyComB)","PAPIIT IN107624"]}],"id":[{"id":"10.13039\/501100005739","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We consider a mathematical model describing the propagation of an epidemic on a geographical network. The size of the outbreak is governed by the initial growth rate of the disease given by the maximal eigenvalue of the epidemic matrix formed by the susceptibles and the graph Laplacian representing the mobility. We use matrix perturbation theory to analyze the epidemic matrix and define a vaccination strategy, assuming vaccination reduces the susceptibles. When mobility and the local disease dynamics have similar time scales, it is most efficient to vaccinate the whole network because the disease grows uniformly. However, if only a few vertices can be vaccinated, then we show that it is most efficient to vaccinate along an eigenvector corresponding to the largest eigenvalue of the Laplacian. We illustrate these results by calculations on a seven-vertex graph and a realistic example of the French rail network. When mobility is slower than the local disease dynamics, the epidemic grows on the vertex with largest number of susceptibles. The epidemic growth rate is more reduced when vaccinating a larger degree vertex; it also depends on the neighboring vertices. This study and its conclusions provide guidelines for the planning of a vaccination campaign on a network at the onset of an epidemic.<\/jats:p>","DOI":"10.3390\/axioms13110769","type":"journal-article","created":{"date-parts":[[2024,11,5]],"date-time":"2024-11-05T11:36:39Z","timestamp":1730806599000},"page":"769","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Mitigating an Epidemic on a Geographic Network Using Vaccination"],"prefix":"10.3390","volume":"13","author":[{"given":"Mohamad","family":"Badaoui","sequence":"first","affiliation":[{"name":"Laboratoire de Math\u00e9matiques, INSA Rouen Normandie, Av. de l\u2019Universit\u00e9, 76801 Saint-Etienne du Rouvray, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8291-6722","authenticated-orcid":false,"given":"Jean-Guy","family":"Caputo","sequence":"additional","affiliation":[{"name":"Laboratoire de Math\u00e9matiques, INSA Rouen Normandie, Av. de l\u2019Universit\u00e9, 76801 Saint-Etienne du Rouvray, France"}]},{"given":"Gustavo","family":"Cruz-Pacheco","sequence":"additional","affiliation":[{"name":"Depto. Instituto de Investigaciones en Matematicas Aplicadas y Sistemas, U.N.A.M., Apdo. Postal 20\u2013726, Ciudad de Mexico 01000, Mexico"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1281-4389","authenticated-orcid":false,"given":"Arnaud","family":"Knippel","sequence":"additional","affiliation":[{"name":"Laboratoire de Math\u00e9matiques, INSA Rouen Normandie, Av. de l\u2019Universit\u00e9, 76801 Saint-Etienne du Rouvray, France"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Latora, V., Nicosia, V., and Russo, G. (2017). Complex Networks: Principles, Methods and Applications, Cambridge University Press.","DOI":"10.1017\/9781316216002"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"056109","DOI":"10.1103\/PhysRevE.65.056109","article-title":"Attack vulnerability of complex networks","volume":"65","author":"Holme","year":"2002","journal-title":"Phys. Rev. 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