{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T08:59:16Z","timestamp":1768467556146,"version":"3.49.0"},"reference-count":38,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,18]],"date-time":"2023-09-18T00:00:00Z","timestamp":1694995200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Youth Talent of Xingdian Talent Support Program","award":["12261104"],"award-info":[{"award-number":["12261104"]}]},{"name":"National Natural Science Foundation of China","award":["12261104"],"award-info":[{"award-number":["12261104"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, we mainly consider the dynamic analysis of a stochastic infectious disease model with negative feedback, a symmetric and compatible distribution family. Based on the sir epidemic model taking into account the isolation (y) and the death (v), we consider adding a new variable (w) to control the information of non-drug interventions, which measures transformations in isolation performance that determine the epidemic, and establish a new model. We have demonstrated various properties of the model solution using Lyapunov functions for this model. To begin with, we demonstrate the existence and uniqueness of the global positive solution. After that, we obtained the conditions that need to be met for the extinction of the disease and verified the correctness of the conclusion by simulating numerical values. Afterwards, we prove the stochastic boundedness and stationary distribution of the model solution.<\/jats:p>","DOI":"10.3390\/sym15091781","type":"journal-article","created":{"date-parts":[[2023,9,18]],"date-time":"2023-09-18T05:59:06Z","timestamp":1695016746000},"page":"1781","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Stochastic Dynamics Analysis of Epidemic Models Considering Negative Feedback of Information"],"prefix":"10.3390","volume":"15","author":[{"given":"Wanqin","family":"Wu","sequence":"first","affiliation":[{"name":"Department of Mathematics, Yunnan Minzu University, 2929, Yuehua Street, Chenggong District, Kunming 650500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wenhui","family":"Luo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Yunnan Minzu University, 2929, Yuehua Street, Chenggong District, Kunming 650500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hui","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Yunnan Minzu University, 2929, Yuehua Street, Chenggong District, Kunming 650500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yun","family":"Zhao","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Yunnan Minzu University, 2929, Yuehua Street, Chenggong District, Kunming 650500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1017\/S0950268800059896","article-title":"Infectious diseases of humans: Dynamics and control","volume":"108","author":"Tillett","year":"1992","journal-title":"Epidemiol. Infect."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/S0022-0396(02)00089-X","article-title":"Dynamical behavior of an epidemic model with a nonlinear incidence rate","volume":"188","author":"Ruan","year":"2003","journal-title":"J. Dierential Equ."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1247","DOI":"10.1098\/rsif.2010.0142","article-title":"Modelling the inuence of human behaviour on the spread of infectious diseases: A review","volume":"7","author":"Funk","year":"2010","journal-title":"J. R. Soc. Interface"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2793","DOI":"10.1090\/S0002-9939-08-09341-6","article-title":"A graph-theoretic approach to the method of global Lyapunov functions","volume":"136","author":"Guo","year":"2018","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2286","DOI":"10.1016\/j.camwa.2010.08.020","article-title":"Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence","volume":"60","author":"Sun","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1677","DOI":"10.1007\/s11071-021-06314-5","article-title":"Mathematical modeling and mechanisms of pattern formation in ecological systems: A review","volume":"104","author":"Sun","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1142\/S1793524521500236","article-title":"Dynamical analysis of an SIS epidemic model with migration and residence time","volume":"14","author":"Liu","year":"2021","journal-title":"Int. J. Biomath."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"582625","DOI":"10.1155\/2015\/582625","article-title":"Dynamical analysis of an SEIT epidemic model with application to ebola virus transmission in Guinea","volume":"2015","author":"Li","year":"2015","journal-title":"Comput. Math. Methods Med."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"064870","DOI":"10.1155\/2007\/64870","article-title":"Analysis of an SIR epidemic model with pulse vaccination and distributed time delay","volume":"2007","author":"Gao","year":"2007","journal-title":"J. Biomed. Biotechnol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.aml.2017.08.002","article-title":"A remark on stationary distribution of a stochastic SIR epidemic model with double saturated rates","volume":"76","author":"Zhang","year":"2018","journal-title":"Appl. Math. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1016\/j.physa.2018.04.022","article-title":"Qualitative study of a stochastic SIS epidemic model with vertical transmission","volume":"505","author":"Zhang","year":"2018","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2020\/6689089","article-title":"Asymptotic behavior of multigroup SEIR model with nonlinear incidence rates under stochastic perturbations","volume":"2020","author":"Wang","year":"2020","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1016\/j.physa.2016.11.077","article-title":"Stationary distribution and extinction of a stochastic SIRS epidemic model with standard incidence","volume":"409","author":"Liu","year":"2017","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1016\/j.aml.2013.11.002","article-title":"The threshold of a stochastic SIRS epidemic model with saturated incidence","volume":"34","author":"Zhao","year":"2014","journal-title":"Appl. Math. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/j.tpb.2007.01.001","article-title":"Vaccinating behaviour, information, and the dynamics of sir vaccine preventable diseases","volume":"71","author":"Manfredi","year":"2007","journal-title":"Theor. Popul. Biol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1019","DOI":"10.3934\/mbe.2017053","article-title":"Global stability of infectious disease models with contact rate as a function of prevalence index","volume":"14","author":"Vargas","year":"2017","journal-title":"Math. Biosci. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.ecocom.2018.06.006","article-title":"Global stability of an information-related epidemic model with age-dependent latency and relapse","volume":"36","author":"Hu","year":"2018","journal-title":"Ecol. Comple"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/j.mbs.2008.07.011","article-title":"Global stability of an sir epidemic model with information dependent vaccination","volume":"216","author":"Buonomo","year":"2008","journal-title":"Math. Biosci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1056","DOI":"10.1016\/j.aml.2012.03.016","article-title":"Globally stable endemicity for infectious diseases with information-related changes in contact patterns","volume":"25","author":"Buonomo","year":"2012","journal-title":"Appl. Math. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1016\/j.jtbi.2003.11.014","article-title":"Simulating the SARS outbreak in Beijing with limited data","volume":"227","author":"Wang","year":"2004","journal-title":"J. Theor. Biol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1055\/s-0030-1265897","article-title":"The role of immunity and inflammation in lung senescence and susceptibility to infection in the elderly","volume":"31","author":"Meyer","year":"2010","journal-title":"Semin. Respir. Crit. Care Med."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.nonrwa.2008.10.014","article-title":"Complete global stability for an SIR epidemic model with delay\u2013Distributed or discrete\u2014ScienceDirect","volume":"11","author":"McCluskey","year":"2010","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"864","DOI":"10.1016\/j.physa.2018.08.048","article-title":"Unique stationary distribution and ergodicity of a stochastic Logistic model with distributed delay","volume":"512","author":"Sun","year":"2018","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_24","unstructured":"Roxana, L.C. (2022). Advances in Continuous and Discrete Models, Springer."},{"key":"ref_25","first-page":"1731","article-title":"Stochastic differential equations: An introduction with applications","volume":"51","author":"ksendal","year":"2006","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_26","unstructured":"Allen, L.J.S. (2008). Mathematical Epidemiology, Springer."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1126\/science.197.4302.463","article-title":"Harvesting natural populations in a randomly fluctuating environment","volume":"197","author":"Beddington","year":"1977","journal-title":"Science"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"121504","DOI":"10.1016\/j.physa.2019.121504","article-title":"A stochastic SIS epidemic model with saturating contact rate","volume":"529","author":"Lan","year":"2019","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Mao, X.R. (2011). Stochastic Differential Equations and Applications, Elsevier.","DOI":"10.1533\/9780857099402.47"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.matcom.2022.08.001","article-title":"Stochastic dynamics of an SIS epidemiological model with media coverage","volume":"204","author":"Tan","year":"2023","journal-title":"Math. Comput. Simul."},{"key":"ref_31","first-page":"1062","article-title":"Classification of asymptotic behavior in a stochastic SIR Model","volume":"15","author":"Dieu","year":"2015","journal-title":"Soc. Ind. Appl. Math."},{"key":"ref_32","first-page":"160","article-title":"A stochastic SIS epidemic model incorporating media coverage in a two patch setting","volume":"262","author":"Liu","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_33","unstructured":"Kiessler, P.C. (2004). Statistical Inference for Ergodic Diffusion Processes, Springer."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1137\/S0036144500378302","article-title":"An algorithmic introduction to numerical simulation of stochastic differential equations","volume":"43","author":"Higham","year":"2001","journal-title":"Siam Rev."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1155\/2013\/172631","article-title":"Stochastic dynamics of a SIRS epidemic model with ratio-dependent incidence rate","volume":"2013","author":"Cai","year":"2013","journal-title":"Abstr. Appl. Anal."},{"key":"ref_36","first-page":"126236","article-title":"Stationary solution, extinction and density function for a high-dimensional stochastic SEI epidemic model with general distributed delay","volume":"405","author":"Han","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"107931","DOI":"10.1016\/j.aml.2022.107931","article-title":"Density function and stationary distribution of a stochastic SIR model with distributed delay","volume":"129","author":"Zuo","year":"2022","journal-title":"Appl. Math. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Khasminskii, R. (2011). Stochastic Stability of Differential Equations, Springer.","DOI":"10.1007\/978-3-642-23280-0"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/9\/1781\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:52:51Z","timestamp":1760129571000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/9\/1781"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,18]]},"references-count":38,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["sym15091781"],"URL":"https:\/\/doi.org\/10.3390\/sym15091781","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,18]]}}}