{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T11:09:30Z","timestamp":1778497770388,"version":"3.51.4"},"reference-count":33,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2025,7,23]],"date-time":"2025-07-23T00:00:00Z","timestamp":1753228800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"name":"MODCOV19"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,7,23]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>During an epidemic, the way individuals spread the infection can vary depending on the number of days they have been infected. Vaccination is a key factor in the control of the transmission of the pathogen. We propose an SIR-like model to simulate the propagation of a virus in a closed population. The model presents a continuous structure in age of infectivity and in age of immunity. By using a fixed point theorem, we then prove the well-posedness of the model and properties of its solution. Then we present the radial basis function-generated finite difference method, which we use to numerically solve our problem. Finally, numerical simulations are explored to witness the impact of the parameter values in the model.<\/jats:p>","DOI":"10.1093\/imamci\/dnaf020","type":"journal-article","created":{"date-parts":[[2025,7,14]],"date-time":"2025-07-14T14:07:08Z","timestamp":1752502028000},"source":"Crossref","is-referenced-by-count":2,"title":["An integro\u2013ODE\u2013PDE model for Covid-19 pandemic with vaccination and loss of immunity: well-posedness, RBF-FD numerical scheme and simulation"],"prefix":"10.1093","volume":"42","author":[{"given":"Robin","family":"Vaudry","sequence":"first","affiliation":[{"name":"Universit\u00e9 Grenoble Alpes, CNRS, Inria, Grenoble INP , LJK, F-38000 Grenoble ,","place":["France"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Candy","family":"Sonveaux","sequence":"additional","affiliation":[{"name":"Namur Institute for Complex Systems (naXys) and Department of Mathematics, University of Namur , Namur ,","place":["Belgium"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Didier","family":"Georges","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Grenoble Alpes, Grenoble INP , GIPSA-lab, F-38000 Grenoble ,","place":["France"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cl\u00e9mentine","family":"Prieur","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Grenoble Alpes, CNRS , Inria, Grenoble INP, LJK, F-38000 Grenoble ,","place":["France"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2025,8,7]]},"reference":[{"key":"2026051106115801900_ref1a","article-title":"Hommage \u00e0 Elisabeta Vergu","author":"Bansaye","year":"2023"},{"key":"2026051106115801900_ref1","doi-asserted-by":"crossref","first-page":"1737","DOI":"10.1007\/s00285-015-0880-5","article-title":"Immuno-epidemiology of a population structured by immune status: a mathematical study of waning immunity and immune system boosting","volume":"71","author":"Barbarossa","year":"2015","journal-title":"J. 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