{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:23:56Z","timestamp":1760059436025,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,13]],"date-time":"2025-06-13T00:00:00Z","timestamp":1749772800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Umm Al-Qura University, Saudi Arabia","award":["25UQU4361068GSSR03"],"award-info":[{"award-number":["25UQU4361068GSSR03"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we study the global dynamics, boundedness, existence of invariant intervals, and identification of codimension-two bifurcation sets with detailed bifurcation analysis at the epidemic fixed point of a discrete epidemic model. More precisely, under definite parametric conditions, it is proved that every positive solution of the discrete epidemic model is bounded, and furthermore, we have also constructed the invariant interval. By the linear stability theory, we have derived the sufficient condition, as well as the necessary and sufficient condition(s) under which fixed points obey certain local dynamical characteristics. We also gave the global analysis at fixed points and proved that both disease-free and epidemic fixed points become globally stable under certain conditions and parameters. Next, in order to study the two-parameter bifurcations of the discrete epidemic model at the epidemic fixed point, we first identified the two-parameter bifurcation sets, and then a detailed two-parameter bifurcation analysis is given by the bifurcation theory and affine transformations. Furthermore, we have given the biological interpretations of the theoretical findings. Finally, numerical simulation validated the theoretical results.<\/jats:p>","DOI":"10.3390\/axioms14060463","type":"journal-article","created":{"date-parts":[[2025,6,13]],"date-time":"2025-06-13T06:19:28Z","timestamp":1749795568000},"page":"463","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Codimension-Two Bifurcation Analysis and Global Dynamics of a Discrete Epidemic Model"],"prefix":"10.3390","volume":"14","author":[{"given":"Raja Ramiz Ahmed","family":"Khan","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad 13100, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0278-1352","authenticated-orcid":false,"given":"Abdul Qadeer","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad 13100, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5322-4978","authenticated-orcid":false,"given":"Turki D.","family":"Alharbi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Al-Leith University College, Umm Al-Qura University, Mecca 24382, Saudi Arabia"}]},{"given":"Jawharah G.","family":"AL-Juaid","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Collage of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Nii-Trebi, N.I. (2017). Emerging and neglected infectious diseases: Insights, advances, and challenges. BioMed Res. Int., 2017.","DOI":"10.1155\/2017\/5245021"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1016\/j.cll.2004.05.008","article-title":"Major factors affecting the emergence and re-emergence of infectious diseases","volume":"24","author":"Church","year":"2004","journal-title":"Clin. Lab. Med."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Frank, S.A. (2002). Immunology and Evolution of Infectious Disease, Princeton University Press.","DOI":"10.1515\/9780691220161"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Li, M.Y. (2018). 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