{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T21:09:51Z","timestamp":1769548191441,"version":"3.49.0"},"reference-count":44,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,13]],"date-time":"2022-08-13T00:00:00Z","timestamp":1660348800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"FCT (Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia)","award":["UIDB\/04106\/2020 (CIDMA)"],"award-info":[{"award-number":["UIDB\/04106\/2020 (CIDMA)"]}]},{"name":"FCT (Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia)","award":["CEECIND\/00564\/2018"],"award-info":[{"award-number":["CEECIND\/00564\/2018"]}]},{"DOI":"10.13039\/501100001871","name":"FCT via the FCT Researcher Program CEEC Individual 2018","doi-asserted-by":"publisher","award":["UIDB\/04106\/2020 (CIDMA)"],"award-info":[{"award-number":["UIDB\/04106\/2020 (CIDMA)"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"FCT via the FCT Researcher Program CEEC Individual 2018","doi-asserted-by":"publisher","award":["CEECIND\/00564\/2018"],"award-info":[{"award-number":["CEECIND\/00564\/2018"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We analyze mathematical models for COVID-19 with discrete time delays and vaccination. Sufficient conditions for the local stability of the endemic and disease-free equilibrium points are proved for any positive time delay. The stability results are illustrated through numerical simulations performed in MATLAB.<\/jats:p>","DOI":"10.3390\/axioms11080400","type":"journal-article","created":{"date-parts":[[2022,8,14]],"date-time":"2022-08-14T21:09:06Z","timestamp":1660511346000},"page":"400","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Stability Analysis of Delayed COVID-19 Models"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2042-7226","authenticated-orcid":false,"given":"Mohamed A.","family":"Zaitri","sequence":"first","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\/0000-0001-8641-2505","authenticated-orcid":false,"given":"Delfim F. M.","family":"Torres","sequence":"additional","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Porta, M. (2014). A Dictionary of Epidemiology, Oxford University Press.","DOI":"10.1093\/acref\/9780199976720.001.0001"},{"key":"ref_2","unstructured":"(2021, December 29). WHO, World Health Organization. Available online: http:\/\/www.emro.who.int\/pandemic-epidemic-diseases\/outbreaks\/index.html."},{"key":"ref_3","unstructured":"(2021, December 29). WHO, World Health Organization. Available online: https:\/\/www.who.int\/health-topics\/hiv-aids."},{"key":"ref_4","unstructured":"(2021, December 29). WHO, World Health Organization. Available online: https:\/\/www.who.int\/health-topics\/tuberculosis."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"105924","DOI":"10.1016\/j.ijantimicag.2020.105924","article-title":"Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and coronavirus disease-2019 (COVID-19): The epidemic and the challenges","volume":"55","author":"Lai","year":"2020","journal-title":"Int. J. Antimicrob. Agents"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"105948","DOI":"10.1016\/j.ijantimicag.2020.105948","article-title":"Review of the 2019 novel coronavirus (SARS-CoV-2) based on current evidence","volume":"55","author":"Wang","year":"2020","journal-title":"Int. J. Antimicrob. Agents"},{"key":"ref_7","unstructured":"WHO, World Health Organization (2021, December 29). WHO Director-General\u2019s Opening Remarks at the Media Briefing on COVID-19. 11 March 2020. Available online: https:\/\/www.who.int\/director-general\/speeches\/detail\/who-director-general-s-opening-remarks-at-the-media-briefing-on-covid-19---11-march-2020."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"e061602","DOI":"10.1136\/bmjopen-2022-061602","article-title":"Questioning the seasonality of SARS-COV-2: A Fourier spectral analysis","volume":"12","author":"Cappi","year":"2022","journal-title":"BMJ Open"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1038\/s41586-021-04130-w","article-title":"Cryptic transmission of SARS-CoV-2 and the first COVID-19 wave","volume":"600","author":"Davis","year":"2021","journal-title":"Nature"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"100885","DOI":"10.1016\/j.ecocom.2020.100885","article-title":"A new compartmental epidemiological model for COVID-19 with a case study of Portugal","volume":"44","author":"Silva","year":"2020","journal-title":"Ecol. Complex."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"110652","DOI":"10.1016\/j.chaos.2021.110652","article-title":"Fractional model of COVID-19 applied to Galicia, Spain and Portugal","volume":"144","author":"Area","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3451","DOI":"10.1038\/s41598-021-83075-6","article-title":"Optimal control of the COVID-19 pandemic: Controlled sanitary deconfinement in Portugal","volume":"11","author":"Silva","year":"2021","journal-title":"Sci. Rep."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"16400","DOI":"10.1038\/s41598-021-95815-9","article-title":"A simple mathematical model for the evaluation of the long first wave of the COVID-19 pandemic in Brazil","volume":"11","author":"Tang","year":"2021","journal-title":"Sci. Rep."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1051\/mmnp\/2020040","article-title":"A stochastic time-delayed model for the effectiveness of Moroccan COVID-19 deconfinement strategy","volume":"15","author":"Zine","year":"2020","journal-title":"Math. Model. Nat. Phenom."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1016\/j.arcontrol.2021.04.005","article-title":"Second special section on systems and control research efforts against COVID-19 and future pandemics","volume":"51","author":"Giordano","year":"2021","journal-title":"Annu. Rev. Control"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Agarwal, P., Nieto, J.J., Ruzhansky, M., and Torres, D.F.M. (2021). Analysis of Infectious Disease Problems (COVID-19) and Their Global Impact, Infosys Science Foundation Series in Mathematical Sciences, Springer.","DOI":"10.1007\/978-981-16-2450-6"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/978-3-030-85053-1_2","article-title":"Describing, modelling and forecasting the spatial and temporal spread of COVID-19: A short review","volume":"Volume 85","author":"Arino","year":"2022","journal-title":"Mathematics of Public Health"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"107783","DOI":"10.1016\/j.aml.2021.107783","article-title":"On the dynamical model for COVID-19 with vaccination and time-delay effects: A model analysis supported by Yangzhou epidemic in 2021","volume":"125","author":"Wang","year":"2022","journal-title":"Appl. Math. Lett."},{"key":"ref_19","first-page":"539","article-title":"Time delays in epidemic models, modeling and numerical considerations","volume":"13","author":"Arino","year":"2006","journal-title":"Delay Differ. Equ. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1002\/oca.2558","article-title":"Optimal control of HIV treatment and immunotherapy combination with state and control delays","volume":"41","author":"Silva","year":"2020","journal-title":"Optim. Control Appl. Meth."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"321","DOI":"10.3934\/mbe.2017021","article-title":"Optimal control of a tuberculosis model with state and control delays","volume":"14","author":"Silva","year":"2017","journal-title":"Math. Biosci. Eng."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1016\/j.chaos.2015.02.017","article-title":"The effect of time delay on the dynamics of an SEIR model with nonlinear incidence","volume":"75","author":"Tipsri","year":"2015","journal-title":"Chaos Solitons Fractals"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1039","DOI":"10.1093\/aje\/kwg251","article-title":"The interval between successive cases of an infectious disease","volume":"158","author":"Fine","year":"2003","journal-title":"Am. J. Epidemiol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"672","DOI":"10.1038\/s41591-020-0869-5","article-title":"Temporal dynamics in viral shedding and transmissibility of COVID-19","volume":"26","author":"He","year":"2020","journal-title":"Nat. Med."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1678","DOI":"10.1093\/cid\/ciab746","article-title":"Estimating the latent period of coronavirus disease 2019 (COVID-19)","volume":"74","author":"Xin","year":"2022","journal-title":"Clin. Infect. Dis."},{"key":"ref_26","unstructured":"WHO, World Health Organization (2021, December 29). Novel Coronavirus (2019-nCoV): Situation Report-7. Available online: https:\/\/apps.who.int\/iris\/handle\/10665\/330771."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"100082","DOI":"10.1016\/j.lanepe.2021.100082","article-title":"Do asymptomatic carriers of SARS-COV-2 transmit the virus?","volume":"4","author":"Muller","year":"2021","journal-title":"Lancet Reg. Health\u2014Europe"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Peng, L., Yang, W., Zhang, D., Zhuge, C., and Hong, L. (2020). Epidemic analysis of COVID-19 in China by dynamical modeling. arXiv.","DOI":"10.1101\/2020.02.16.20023465"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1007\/s00285-021-01657-4","article-title":"Continuous-time stochastic processes for the spread of COVID-19 disease simulated via a Monte Carlo approach and comparison with deterministic models","volume":"83","author":"Calleri","year":"2021","journal-title":"J. Math. Biol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1186\/s13662-020-02964-8","article-title":"Stochastic SIRC epidemic model with time-delay for COVID-19","volume":"2020","author":"Rihan","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Zaitri, M.A., Bibi, M.O., and Torres, D.F.M. (2021). Optimal control to limit the spread of COVID-19 in Italy. Kuwait J. Sci., 1\u201314. Special Issue on COVID.","DOI":"10.48129\/kjs.splcov.13961"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Dong, Y., Kourtellis, N., Hammer, B., and Lozano, J.A. (2021). Open Data Science to Fight COVID-19: Winning the 500k XPRIZE Pandemic Response Challenge. Machine Learning and Knowledge Discovery in Databases. Applied Data Science Track. ECML PKDD 2021. Lecture Notes in Computer Science, Springer.","DOI":"10.1007\/978-3-030-86517-7"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1109\/TEVC.2021.3063217","article-title":"From Prediction to Prescription: Evolutionary Optimization of Nonpharmaceutical Interventions in the COVID-19 Pandemic","volume":"25","author":"Miikkulainen","year":"2021","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"855","DOI":"10.1038\/s41591-020-0883-7","article-title":"Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy","volume":"26","author":"Giordano","year":"2020","journal-title":"Nat. Med."},{"key":"ref_35","first-page":"323","article-title":"A COVID-19 epidemic model with latency period","volume":"5","author":"Liu","year":"2020","journal-title":"Infect. Dis. Model."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/S0025-5564(02)00108-6","article-title":"Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission","volume":"180","author":"Watmough","year":"2002","journal-title":"Math. Biosci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"448","DOI":"10.3934\/Math.2018.4.448","article-title":"A cholera mathematical model with vaccination and the biggest outbreak of world\u2019s history","volume":"3","author":"Silva","year":"2018","journal-title":"AIMS Math."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1137\/1025075","article-title":"Locations of roots of polynomials","volume":"25","author":"Rogers","year":"1983","journal-title":"SIAM Rev."},{"key":"ref_39","unstructured":"Kuang, Y. (1993). Delay Differential Equations with Applications in Population Dynamics, Academic Press, Inc.. Mathematics in Science and Engineering, 191."},{"key":"ref_40","unstructured":"Niculescu, S.-I. (2001). Delay Effects on Stability, Springer. Lecture Notes in Control and Information Sciences, 269."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1137\/S1064827594276424","article-title":"The MATLAB ODE suite","volume":"18","author":"Shampine","year":"1997","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"125171","DOI":"10.1016\/j.jmaa.2021.125171","article-title":"Complex network model for COVID-19: Human behavior, pseudo-periodic solutions and multiple epidemic waves","volume":"514","author":"Silva","year":"2022","journal-title":"J. Math. Anal. Appl."},{"key":"ref_43","unstructured":"United Nations (2021, December 29). The 2022 Revision of World Population Prospects. Available online: https:\/\/population.un.org\/wpp\/."},{"key":"ref_44","unstructured":"Cheynet, E. (2020). Generalized SEIR Epidemic Model (Fitting and Computation), Zenodo. Transform to Open Science, NASA."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/8\/400\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:08:16Z","timestamp":1760141296000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/8\/400"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,13]]},"references-count":44,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["axioms11080400"],"URL":"https:\/\/doi.org\/10.3390\/axioms11080400","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,8,13]]}}}