{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,22]],"date-time":"2026-03-22T07:44:20Z","timestamp":1774165460536,"version":"3.50.1"},"reference-count":55,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,10,10]],"date-time":"2024-10-10T00:00:00Z","timestamp":1728518400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11361104"],"award-info":[{"award-number":["11361104"]}]},{"name":"National Natural Science Foundation of China","award":["12261104"],"award-info":[{"award-number":["12261104"]}]},{"name":"National Natural Science Foundation of China","award":["XDYC-QNRC-2022-0514"],"award-info":[{"award-number":["XDYC-QNRC-2022-0514"]}]},{"name":"National Natural Science Foundation of China","award":["202301AT070016"],"award-info":[{"award-number":["202301AT070016"]}]},{"name":"National Natural Science Foundation of China","award":["202401AT070036"],"award-info":[{"award-number":["202401AT070036"]}]},{"name":"National Natural Science Foundation of China","award":["2024Y468"],"award-info":[{"award-number":["2024Y468"]}]},{"name":"Youth Talent Program of Xingdian Talent Support Plan","award":["11361104"],"award-info":[{"award-number":["11361104"]}]},{"name":"Youth Talent Program of Xingdian Talent Support Plan","award":["12261104"],"award-info":[{"award-number":["12261104"]}]},{"name":"Youth Talent Program of Xingdian Talent Support Plan","award":["XDYC-QNRC-2022-0514"],"award-info":[{"award-number":["XDYC-QNRC-2022-0514"]}]},{"name":"Youth Talent Program of Xingdian Talent Support Plan","award":["202301AT070016"],"award-info":[{"award-number":["202301AT070016"]}]},{"name":"Youth Talent Program of Xingdian Talent Support Plan","award":["202401AT070036"],"award-info":[{"award-number":["202401AT070036"]}]},{"name":"Youth Talent Program of Xingdian Talent Support Plan","award":["2024Y468"],"award-info":[{"award-number":["2024Y468"]}]},{"name":"Yunnan Provincial Basic Research Program Project","award":["11361104"],"award-info":[{"award-number":["11361104"]}]},{"name":"Yunnan Provincial Basic Research Program Project","award":["12261104"],"award-info":[{"award-number":["12261104"]}]},{"name":"Yunnan Provincial Basic Research Program Project","award":["XDYC-QNRC-2022-0514"],"award-info":[{"award-number":["XDYC-QNRC-2022-0514"]}]},{"name":"Yunnan Provincial Basic Research Program Project","award":["202301AT070016"],"award-info":[{"award-number":["202301AT070016"]}]},{"name":"Yunnan Provincial Basic Research Program Project","award":["202401AT070036"],"award-info":[{"award-number":["202401AT070036"]}]},{"name":"Yunnan Provincial Basic Research Program Project","award":["2024Y468"],"award-info":[{"award-number":["2024Y468"]}]},{"name":"Science Research Fund of Education Department of Yunnan Province","award":["11361104"],"award-info":[{"award-number":["11361104"]}]},{"name":"Science Research Fund of Education Department of Yunnan Province","award":["12261104"],"award-info":[{"award-number":["12261104"]}]},{"name":"Science Research Fund of Education Department of Yunnan Province","award":["XDYC-QNRC-2022-0514"],"award-info":[{"award-number":["XDYC-QNRC-2022-0514"]}]},{"name":"Science Research Fund of Education Department of Yunnan Province","award":["202301AT070016"],"award-info":[{"award-number":["202301AT070016"]}]},{"name":"Science Research Fund of Education Department of Yunnan Province","award":["202401AT070036"],"award-info":[{"award-number":["202401AT070036"]}]},{"name":"Science Research Fund of Education Department of Yunnan Province","award":["2024Y468"],"award-info":[{"award-number":["2024Y468"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This study investigates a symmetric fractional-order epidemic model with time delays and non-monotonic incidence rates, considering two viral strains. By confirming the existence, uniqueness, and boundedness of the system\u2019s solutions, the research ensures the model\u2019s well-posedness, guaranteeing its mathematical soundness and practical relevance. The study calculates and evaluates the equilibrium points and the basic reproduction numbers R01 and R02 to understand the dynamic behavior of the model under different parameter settings. Through the application of the Lyapunov method, the research examines the asymptotic global stability of the system, determining whether it will converge to a particular equilibrium state over time. Furthermore, Hopf bifurcation theory is employed to investigate potential periodic solutions and bifurcation scenarios, highlighting how the system might shift from stability to periodic oscillations under certain conditions. By utilizing the Adams-Bashforth-Moulton numerical simulation method, the theoretical results are validated, reinforcing the conclusions and demonstrating the model\u2019s applicability in real-world contexts. It emphasizes the importance of fractional-order models in addressing epidemiological issues related to time delays (\u03c4), individual heterogeneity (m, k), and memory effects (\u03b8), offering greater accuracy compared with traditional integer-order models. In summary, this research provides a theoretical foundation and practical insights, enhancing the understanding and management of epidemic dynamics through fractional-order epidemic models.<\/jats:p>","DOI":"10.3390\/sym16101343","type":"journal-article","created":{"date-parts":[[2024,10,10]],"date-time":"2024-10-10T11:34:36Z","timestamp":1728560076000},"page":"1343","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Stability and Bifurcation Analysis of a Symmetric Fractional-Order Epidemic Mathematical Model with Time Delay and Non-Monotonic Incidence Rates for Two Viral Strains"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0009-0004-1528-5210","authenticated-orcid":false,"given":"Zhixiang","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematics and Computer Science, Yunnan Minzu University, 2929, Yuehua Street, Chenggong District, Kunming 650500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wanqin","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Yunnan Minzu University, 2929, Yuehua Street, Chenggong District, Kunming 650500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xuewen","family":"Tan","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Yunnan Minzu University, 2929, Yuehua Street, Chenggong District, Kunming 650500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qing","family":"Miao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Yunnan Minzu University, 2929, Yuehua Street, Chenggong District, Kunming 650500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,10]]},"reference":[{"key":"ref_1","first-page":"700","article-title":"A contribution to the mathematical theory of epidemics","volume":"115","author":"Kermack","year":"1927","journal-title":"Proc. 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