{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T04:34:23Z","timestamp":1775190863400,"version":"3.50.1"},"reference-count":53,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,7]],"date-time":"2022-07-07T00:00:00Z","timestamp":1657152000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Spanish Government","award":["RTI2018-094336-B-I00"],"award-info":[{"award-number":["RTI2018-094336-B-I00"]}]},{"name":"Spanish Government","award":["IT1207-19"],"award-info":[{"award-number":["IT1207-19"]}]},{"name":"Basque Government","award":["RTI2018-094336-B-I00"],"award-info":[{"award-number":["RTI2018-094336-B-I00"]}]},{"name":"Basque Government","award":["IT1207-19"],"award-info":[{"award-number":["IT1207-19"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This manuscript deals with the qualitative study of certain properties of an immunogenic tumors model. Mainly, we obtain a dynamically consistent discrete-time immunogenic tumors model using a nonstandard difference scheme. The existence of fixed points and their stability are discussed. It is shown that a continuous system experiences Hopf bifurcation at one and only one positive fixed point, whereas its discrete-time counterpart experiences Neimark\u2013Sacker bifurcation at one and only one positive fixed point. It is shown that there is no chance of period-doubling bifurcation in our discrete-time system. Additionally, numerical simulations are carried out in support of our theoretical discussion.<\/jats:p>","DOI":"10.3390\/e24070949","type":"journal-article","created":{"date-parts":[[2022,7,7]],"date-time":"2022-07-07T22:11:47Z","timestamp":1657231907000},"page":"949","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Dynamically Consistent Nonstandard Difference Scheme for a Discrete-Time Immunogenic Tumors Model"],"prefix":"10.3390","volume":"24","author":[{"given":"Muhammad Salman","family":"Khan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Quaid-I-Azam University, Islamabad 44230, Pakistan"}]},{"given":"Maria","family":"Samreen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Quaid-I-Azam University, Islamabad 44230, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6316-5230","authenticated-orcid":false,"given":"Muhammad Asif","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Kahota-Haveli Campus, University of the Poonch Rawalakot, Rawalakot 12350, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9320-9433","authenticated-orcid":false,"given":"Manuel","family":"De la Sen","sequence":"additional","affiliation":[{"name":"Department of Electricity and Electronics, Institute of Research and Development of Processes, Faculty of Science and Technology, University of the Basque Country (UPV\/EHU), Campus of Leioa, 48940 Leioa, Bizkaia, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1051\/mmnp\/2019002","article-title":"Fractional order model of immune cells influenced by cancer cells","volume":"14","author":"Ucar","year":"2019","journal-title":"Math. 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