{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T10:31:38Z","timestamp":1760783498308,"version":"3.41.2"},"reference-count":16,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2012,1,30]],"date-time":"2012-01-30T00:00:00Z","timestamp":1327881600000},"content-version":"vor","delay-in-days":29,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100004480","name":"Natural Science Foundation of Shanxi Province","doi-asserted-by":"publisher","award":["2008011002-2","210030"],"award-info":[{"award-number":["2008011002-2","210030"]}],"id":[{"id":"10.13039\/501100004480","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002338","name":"Ministry of Education of the People\u2019s Republic of China","doi-asserted-by":"publisher","award":["2008011002-2","210030"],"award-info":[{"award-number":["2008011002-2","210030"]}],"id":[{"id":"10.13039\/501100002338","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>We formulate and discuss models for the spread of infectious diseases with variable\npopulation sizes and vaccinations on the susceptible individuals. First, we assume that the susceptible individuals are vaccinated\ncontinuously. We establish the threshold\u2010like results for the existence and global stability of the disease\u2010free and the endemic\nequilibriums for these systems. Especially, we prove the global stability of the endemic equilibriums by converting the systems into\nintegrodifferential equations. Second, we suppose that vaccinations occur once per time period. We obtain the existence and global\nstability of the disease\u2010free periodic solutions for such systems with impulsive effects. By a useful bifurcation theorem, we acquire\nthe existence of the endemic periodic solutions when the disease\u2010related deaths do not occur. At last, we compare the\nresults with vaccinations and without vaccinations and illustrate our results by numerical simulations.<\/jats:p>","DOI":"10.1155\/2012\/824192","type":"journal-article","created":{"date-parts":[[2012,1,30]],"date-time":"2012-01-30T21:01:15Z","timestamp":1327957275000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Dynamical Models for Infectious Diseases with Varying Population Size and Vaccinations"],"prefix":"10.1155","volume":"2012","author":[{"given":"Peilin","family":"Shi","sequence":"first","affiliation":[]},{"given":"Lingzhen","family":"Dong","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,1,30]]},"reference":[{"key":"e_1_2_6_1_2","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144500371907"},{"key":"e_1_2_6_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF00173264"},{"key":"e_1_2_6_3_2","doi-asserted-by":"publisher","DOI":"10.1038\/280361a0"},{"key":"e_1_2_6_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF00173265"},{"key":"e_1_2_6_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0025-5564(99)00030-9"},{"key":"e_1_2_6_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0025-5564(01)00057-8"},{"key":"e_1_2_6_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0092-8240(98)90005-2"},{"key":"e_1_2_6_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0895-7177(00)00040-6"},{"key":"e_1_2_6_9_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0895-7177(02)00082-1"},{"key":"e_1_2_6_10_2","doi-asserted-by":"publisher","DOI":"10.1006\/tpbi.1995.1006"},{"key":"e_1_2_6_11_2","doi-asserted-by":"publisher","DOI":"10.1093\/imammb\/9.2.67"},{"key":"e_1_2_6_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0025-5564(97)10016-5"},{"key":"e_1_2_6_13_2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511662805"},{"key":"e_1_2_6_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(76)90096-7"},{"key":"e_1_2_6_15_2","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(65)90126-5"},{"key":"e_1_2_6_16_2","first-page":"265","article-title":"Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment","volume":"7","author":"Lakmeche A.","year":"2000","journal-title":"Dynamics of Continuous, Discrete and Impulsive Systems"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2012\/824192.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2012\/824192.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2012\/824192","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,11]],"date-time":"2024-06-11T07:19:56Z","timestamp":1718090396000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2012\/824192"}},"subtitle":[],"editor":[{"given":"Venky","family":"Krishnan","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2012,1]]},"references-count":16,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2012,1]]}},"alternative-id":["10.1155\/2012\/824192"],"URL":"https:\/\/doi.org\/10.1155\/2012\/824192","archive":["Portico"],"relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2012,1]]},"assertion":[{"value":"2011-07-24","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2011-10-13","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-01-30","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"824192"}}