{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,20]],"date-time":"2026-05-20T03:16:00Z","timestamp":1779246960488,"version":"3.51.4"},"reference-count":53,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,4,11]],"date-time":"2024-04-11T00:00:00Z","timestamp":1712793600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Shaanxi Province","award":["2023-JC-YB-083"],"award-info":[{"award-number":["2023-JC-YB-083"]}]},{"name":"Natural Science Foundation of Shaanxi Province","award":["12361041"],"award-info":[{"award-number":["12361041"]}]},{"name":"Natural Science Foundation of Shaanxi Province","award":["12061033"],"award-info":[{"award-number":["12061033"]}]},{"name":"Natural Science Foundation of Shaanxi Province","award":["12201499"],"award-info":[{"award-number":["12201499"]}]},{"name":"National Natural Science Foundation of China","award":["2023-JC-YB-083"],"award-info":[{"award-number":["2023-JC-YB-083"]}]},{"name":"National Natural Science Foundation of China","award":["12361041"],"award-info":[{"award-number":["12361041"]}]},{"name":"National Natural Science Foundation of China","award":["12061033"],"award-info":[{"award-number":["12061033"]}]},{"name":"National Natural Science Foundation of China","award":["12201499"],"award-info":[{"award-number":["12201499"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper delves into the analysis of a stochastic epidemic model known as the susceptible\u2013vaccinated\u2013exposed\u2013infectious\u2013recovered (SVEIR) model, where transmission dynamics are governed by a nonlinear function. In the theoretical analysis section, by suitable stochastic Lyapunov functions, we establish that when the threshold value, denoted as R0s, falls below 1, the epidemic is destined for extinction. Conversely, if the reproduction number R0 of the deterministic model surpasses 1, the model manifests an ergodic endemic stationary distribution. In the numerical simulations and data interpretation section, leveraging a graphical analysis with COVID-19 data, we illustrate that random fluctuations possess the capacity to quell disease outbreaks, underscoring the role of vaccines in curtailing the spread of diseases. This study not only contributes to the understanding of epidemic dynamics but also highlights the pivotal role of stochasticity and vaccination strategies in epidemic control and management. The inherent balance and patterns observed in epidemic spread and control strategies, reflect a symmetrical interplay between stochasticity, vaccination, and disease dynamics.<\/jats:p>","DOI":"10.3390\/sym16040467","type":"journal-article","created":{"date-parts":[[2024,4,12]],"date-time":"2024-04-12T03:34:37Z","timestamp":1712892877000},"page":"467","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Dynamics of a Stochastic SVEIR Epidemic Model with Nonlinear Incidence Rate"],"prefix":"10.3390","volume":"16","author":[{"given":"Xinghao","family":"Wang","sequence":"first","affiliation":[{"name":"College of Science, Northwest A&F University, Yangling 712100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Liang","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Science, Northwest A&F University, Yangling 712100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiao-Bing","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,11]]},"reference":[{"key":"ref_1","first-page":"480","article-title":"Numerical study of SARS epidemic model with the inclusion of diffusion in the system","volume":"229","author":"Naheed","year":"2014","journal-title":"Appl. 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