{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T00:02:46Z","timestamp":1770681766639,"version":"3.49.0"},"reference-count":24,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,12,4]],"date-time":"2024-12-04T00:00:00Z","timestamp":1733270400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11701074"],"award-info":[{"award-number":["11701074"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, a new numerical scheme, which we call the impulsive linearly implicit Euler method, for the SIR epidemic model with pulse vaccination strategy is constructed based on the linearly implicit Euler method. The sufficient conditions for global attractivity of an infection-free periodic solution of the impulsive linearly implicit Euler method are obtained. We further show that the limit of the disease-free periodic solution of the impulsive linearly implicit Euler method is the disease-free periodic solution of the exact solution when the step size tends to 0. Finally, two numerical experiments are given to confirm the conclusions.<\/jats:p>","DOI":"10.3390\/axioms13120854","type":"journal-article","created":{"date-parts":[[2024,12,4]],"date-time":"2024-12-04T10:07:10Z","timestamp":1733306830000},"page":"854","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Impulsive Linearly Implicit Euler Method for the SIR Epidemic Model with Pulse Vaccination Strategy"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6918-4993","authenticated-orcid":false,"given":"Gui-Lai","family":"Zhang","sequence":"first","affiliation":[{"name":"College of Sciences, Northeastern University, Shenyang 110819, China"}]},{"given":"Zhi-Yong","family":"Zhu","sequence":"additional","affiliation":[{"name":"College of Sciences, Northeastern University, Shenyang 110819, China"}]},{"given":"Lei-Ke","family":"Chen","sequence":"additional","affiliation":[{"name":"College of Sciences, Northeastern University, Shenyang 110819, China"}]},{"given":"Song-Shu","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Sciences, Northeastern University, Shenyang 110819, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,4]]},"reference":[{"key":"ref_1","first-page":"700","article-title":"Contributions to the mathematical theory of epidemics","volume":"115","author":"Kermack","year":"1927","journal-title":"Proc. R. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Anderson, R.M., and May, R.M. (1991). Infectious Diseases of Humans: Dynamics and Control, Oxford University Press.","DOI":"10.1093\/oso\/9780198545996.001.0001"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Ma, Z.E., Zhou, Y.C., and Wu, J.H. (2009). Modeling and Dynamics of Infectious Diseases, Higher Education Press.","DOI":"10.1142\/7223"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Brauer, F., Castillo-Chavez, C., and Feng, Z. (2019). Mathematical Models in Epidemiology, Springer.","DOI":"10.1007\/978-1-4939-9828-9"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"11698","DOI":"10.1073\/pnas.90.24.11698","article-title":"Pulse mass measles vaccination across age cohorts","volume":"90","author":"Agur","year":"1993","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1123","DOI":"10.1016\/S0092-8240(98)90005-2","article-title":"Pulse vaccination strategy in the SIR epidemic model","volume":"60","author":"Shulgin","year":"1998","journal-title":"Bull. Math. Biol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/S0895-7177(00)00040-6","article-title":"Theoretical examination of the pulse vaccination policy in the SIR epidemic model","volume":"31","author":"Stone","year":"2000","journal-title":"Math. Comput. Model."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1016\/j.na.2005.05.029","article-title":"New modelling approach concerning integrated disease control and cost-effectivity","volume":"63","author":"Tang","year":"2005","journal-title":"Nonlinear Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"6037","DOI":"10.1016\/j.vaccine.2006.05.018","article-title":"Analysis of a delayed epidemic model with pulse vaccination and saturation incidence","volume":"24","author":"Gao","year":"2006","journal-title":"Vaccine"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1004","DOI":"10.1016\/j.chaos.2007.08.056","article-title":"Analysis of a delayed SIR epidemic model with pulse vaccination","volume":"40","author":"Gao","year":"2009","journal-title":"Chaos Solitons Fractals"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1007\/s12190-009-0321-y","article-title":"The differential susceptibility SIR epidemic model with time delay and pulse vaccination","volume":"34","author":"Zhang","year":"2010","journal-title":"J. Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1137\/S0036139999359860","article-title":"Global dynamics of an SEIR epidemic model with vertical transmission","volume":"62","author":"Li","year":"2001","journal-title":"SIAM J. Appl. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"6281","DOI":"10.3934\/math.2024306","article-title":"Amathematical model for predicting and controlling COVID-19 transmission with impulsive vaccination","volume":"9","author":"Rattanakul","year":"2014","journal-title":"AIMS Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"108272","DOI":"10.1016\/j.cnsns.2024.108272","article-title":"Pulse vaccination in a SIR model: Global dynamics, bifurcations and seasonality","volume":"139","author":"Rodrigues","year":"2024","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"662","DOI":"10.1137\/0523034","article-title":"Analytical and numerical results for the agestructured SIS epidemic model with mixed inter-intracohort transmission","volume":"23","author":"Iannelli","year":"1992","journal-title":"SIAM J. Math. Anal."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/j.apnum.2020.12.018","article-title":"Numerical analysis of linearly implicit Euler\u2013Riemann method for nonlinear Gurtin-MacCamy model","volume":"163","author":"Yang","year":"2021","journal-title":"Appl. Numer. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"70","DOI":"10.3934\/dcdsb.2022067","article-title":"Numerical threshold of linearly implicit Euler method for nonlinear infection-age SIR models","volume":"28","author":"Yang","year":"2023","journal-title":"Discrete Contin. Dyn. Syst. Ser. B"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.matcom.2022.07.021","article-title":"Numerical representations of global epidemic threshold for nonlinear infection-age SIR models","volume":"204","author":"Cao","year":"2023","journal-title":"MaThemat. Comput. Simul."},{"key":"ref_19","unstructured":"Bainov, D.D., and Simeonov, P.S. (1989). Systems with Impulsive Effect: Stability, Theory and Applications, Ellis Horwood."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Bainov, D.D., and Simeonov, P.S. (1995). Impulsive Differential Equations: Asymptotic Properties of the Solutions, World Scientific.","DOI":"10.1142\/9789812831804"},{"key":"ref_21","unstructured":"Ma, Z.E., Zhou, Y.C., and Li, C.Z. (2015). Qualitative and Stability Methods for Ordinary Differential Equations, Science Press. (In Chinese)."},{"key":"ref_22","unstructured":"Hairer, E., N\u00f8rsett, S.P., and Wanner, G. (1993). Solving Ordinary Differential Equations I: Nonstiff Problems, Springer."},{"key":"ref_23","first-page":"127017","article-title":"Convergence, consistency and zero stability of impulsive one-step numerical methods","volume":"423","author":"Zhang","year":"2022","journal-title":"Appl. Math. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"997","DOI":"10.1016\/j.cam.2011.05.040","article-title":"Dynamics of a discretized SIR epidemic model with pulse vaccinationand time delay","volume":"236","author":"Sekiguchia","year":"2011","journal-title":"J. Comput. Appl. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/12\/854\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:46:49Z","timestamp":1760114809000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/12\/854"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,4]]},"references-count":24,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["axioms13120854"],"URL":"https:\/\/doi.org\/10.3390\/axioms13120854","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,12,4]]}}}