{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,17]],"date-time":"2026-02-17T11:32:46Z","timestamp":1771327966169,"version":"3.50.1"},"reference-count":30,"publisher":"SAGE Publications","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IFS"],"published-print":{"date-parts":[[2023,11,4]]},"abstract":"<jats:p>This paper mainly discusses the extinction and persistent dynamic behavior of infectious diseases with temporary immunity. Considering that the transmission process of infectious diseases is affected by environmental fluctuations, stochastic SIRS models have been proposed, while the outbreak of diseases is sudden and the interference terms that affect disease transmission cannot be qualified as random variables. Liu process is introduced based on uncertainty theory, which is a new branch of mathematics for describing uncertainty phenomena, to describe uncertain disturbances in epidemic transmission. This paper first extends the classic SIRS model from a deterministic framework to an uncertain framework and constructs an uncertain SIRS infectious disease model with constant input and bilinear incidence. Then, by means of Yao-Chen formula, \u03b1-path of uncertain SIRS model and the corresponding ordinary differential equations are obtained to introduce the uncertainty threshold function R 0 * as the basic reproduction number. Moreover, two equilibrium states are derived. A series of numerical examples show that the larger the value of R 0 * , the more difficult it is to control the disease. If R 0 * \u2264 1 , the infectious disease will gradually disappear, while if R 0 * &gt; 1 , the infectious disease will develop into a local epidemic.<\/jats:p>","DOI":"10.3233\/jifs-223439","type":"journal-article","created":{"date-parts":[[2023,5,12]],"date-time":"2023-05-12T13:37:48Z","timestamp":1683898668000},"page":"9083-9093","source":"Crossref","is-referenced-by-count":5,"title":["Threshold dynamics of an uncertain SIRS epidemic model with a bilinear incidence"],"prefix":"10.1177","volume":"45","author":[{"given":"Simin","family":"Tan","sequence":"first","affiliation":[{"name":"College of Mathematics and System Science, Xinjiang University, Urumqi, People\u2019s Republic of China"}]},{"given":"Ling","family":"Zhang","sequence":"additional","affiliation":[{"name":"Center for Disease Control and Prevention of Xinjiang Uygur Autonomous Region, Urumqi, People\u2019s Republic of China"}]},{"given":"Yuhong","family":"Sheng","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Science, Xinjiang University, Urumqi, People\u2019s Republic of China"}]}],"member":"179","reference":[{"issue":"772","key":"10.3233\/JIFS-223439_ref1","first-page":"700","article-title":"A contribution to the mathematical theory of epidemics","volume":"115","author":"Kermack","year":"1927","journal-title":"Proceedings of the Royal Society A"},{"key":"10.3233\/JIFS-223439_ref2","first-page":"55","article-title":"Contributions to the mathematical theory of epidemics","volume":"138","author":"Kermack","year":"1932","journal-title":"Proceedings of the Royal Society, A"},{"issue":"1","key":"10.3233\/JIFS-223439_ref3","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/0025-5564(94)90025-6","article-title":"Some discrete-time SI, SIR, and SIS epidemic models","volume":"124","author":"Allen","year":"1994","journal-title":"Mathematical Biosciences"},{"issue":"1","key":"10.3233\/JIFS-223439_ref4","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/S0025-5564(03)00087-7","article-title":"Global dynamics of an SEIR epidemic model with saturating contact rate","volume":"185","author":"Zhang","year":"2003","journal-title":"Mathematical Biosciences"},{"issue":"2","key":"10.3233\/JIFS-223439_ref5","doi-asserted-by":"crossref","first-page":"1058","DOI":"10.1016\/j.apm.2007.12.020","article-title":"Permanence and extinction for a nonautonomous SIRS epidemic model with time delay","volume":"32","author":"Zhang","year":"2009","journal-title":"Applied Mathematical Modelling"},{"issue":"18","key":"10.3233\/JIFS-223439_ref6","doi-asserted-by":"crossref","first-page":"9321","DOI":"10.1016\/j.amc.2012.03.011","article-title":"Permanence and extinction for a nonautonomous SEIRS epidemic model","volume":"218","author":"Kuniya","year":"2012","journal-title":"Applied Mathematics and Computation"},{"issue":"2","key":"10.3233\/JIFS-223439_ref7","doi-asserted-by":"crossref","first-page":"719","DOI":"10.1016\/j.jmaa.2013.08.024","article-title":"Complete global analysis of an SIRS epidemic model with graded cure and incomplete recovery rates","volume":"410","author":"Muroya","year":"2014","journal-title":"Journal of Mathematical Analysis and Applications"},{"key":"10.3233\/JIFS-223439_ref8","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/j.physa.2018.01.007","article-title":"Dynamics and optimal control of a non-linear epidemic model with relapse and cure","volume":"496","author":"Lahrouz","year":"2018","journal-title":"Physica A: Statistical Mechanics and its Applications"},{"issue":"3","key":"10.3233\/JIFS-223439_ref9","doi-asserted-by":"crossref","first-page":"876","DOI":"10.1137\/10081856X","article-title":"A stochastic differential equation SIS epidemic model","volume":"71","author":"Gray","year":"2011","journal-title":"SIAM Journal on Applied Mathematics"},{"issue":"1","key":"10.3233\/JIFS-223439_ref10","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/j.automatica.2011.09.044","article-title":"Dynamics of a multigroup SIR epidemic model with stochastic perturbation","volume":"48","author":"Ji","year":"2012","journal-title":"Automatica"},{"key":"10.3233\/JIFS-223439_ref11","doi-asserted-by":"crossref","first-page":"120696","DOI":"10.1016\/j.physa.2019.03.061","article-title":"Stationary distribution and threshold dynamics of a stochastic SIRS model with a general incidence","volume":"534","author":"El Fatini","year":"2019","journal-title":"Physica A: Statistical Mechanics and its Applications"},{"issue":"12","key":"10.3233\/JIFS-223439_ref12","doi-asserted-by":"crossref","first-page":"6131","DOI":"10.3934\/dcdsb.2021010","article-title":"Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination","volume":"26","author":"Zhao","year":"2021","journal-title":"Discrete & Continuous Dynamical Systems-B"},{"issue":"3","key":"10.3233\/JIFS-223439_ref13","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Information and Control"},{"key":"10.3233\/JIFS-223439_ref14","first-page":"1","article-title":"Dynamic updating approximations of local generalized multigranulation neighborhood rough set","volume":"2022","author":"Xu","year":"2022","journal-title":"Applied Intelligence"},{"key":"10.3233\/JIFS-223439_ref15","first-page":"1","article-title":"Two-way concept-cognitive learning method: A fuzzy-based progressive learning","volume":"2022","author":"Xu","year":"2022","journal-title":"IEEE Transactions on Fuzzy Systems"},{"key":"10.3233\/JIFS-223439_ref16","unstructured":"Liu B. , Uncertainty Theory. Springer-Verlag, Berlin, 2nd edition, 2007."},{"issue":"1","key":"10.3233\/JIFS-223439_ref17","first-page":"3","article-title":"Fuzzy process, hybrid process and uncertain process","volume":"2","author":"Liu","year":"2008","journal-title":"Journal of Uncertain Systems"},{"issue":"1","key":"10.3233\/JIFS-223439_ref18","first-page":"3","article-title":"Some research problems in uncertainty theory","volume":"3","author":"Liu","year":"2009","journal-title":"Journal of Uncertain Systems"},{"key":"10.3233\/JIFS-223439_ref19","unstructured":"Liu B. , Uncertainty Theory. Springer-Verlag, Berlin, 4th edition, 2010."},{"issue":"1","key":"10.3233\/JIFS-223439_ref20","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/s10700-010-9073-2","article-title":"Existence and uniqueness theorem for uncertain differential equations","volume":"9","author":"Chen","year":"2010","journal-title":"Fuzzy Optimization and Decision Making"},{"issue":"3","key":"10.3233\/JIFS-223439_ref21","doi-asserted-by":"crossref","first-page":"825","DOI":"10.3233\/IFS-120688","article-title":"A numerical method for solving uncertain differential equations","volume":"25","author":"Yao","year":"2013","journal-title":"Journal of Intelligent & Fuzzy Systems"},{"issue":"4","key":"10.3233\/JIFS-223439_ref22","doi-asserted-by":"crossref","first-page":"2317","DOI":"10.3233\/JIFS-17354","article-title":"An uncertain differential equation for SIS epidemic model","volume":"33","author":"Li","year":"2017","journal-title":"Journal of Intelligent & Fuzzy Systems"},{"issue":"1","key":"10.3233\/JIFS-223439_ref23","doi-asserted-by":"crossref","first-page":"927","DOI":"10.3233\/JIFS-171684","article-title":"Solution and \u03b1-path of uncertain SIS epidemic model with standard incidence and demography","volume":"35","author":"Li","year":"2018","journal-title":"Journal of Intelligent & Fuzzy Systems"},{"issue":"5","key":"10.3233\/JIFS-223439_ref24","doi-asserted-by":"crossref","first-page":"5785","DOI":"10.3233\/JIFS-18007","article-title":"Comparison of three SIS epidemic models: Deterministic, stochastic and uncertain","volume":"35","author":"Li","year":"2018","journal-title":"Journal of Intelligent & Fuzzy Systems"},{"key":"10.3233\/JIFS-223439_ref25","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1007\/s10700-020-09341-w","article-title":"Uncertain SEIAR model for COVID-19 cases in China","volume":"20","author":"Jia","year":"2021","journal-title":"Fuzzy Optimization and Decision Making"},{"issue":"2","key":"10.3233\/JIFS-223439_ref26","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/s10700-020-09340-x","article-title":"Uncertain growth model for the cumulative number of COVID-19 infections in China","volume":"20","author":"Liu","year":"2021","journal-title":"Fuzzy Optimization and Decision Making"},{"issue":"2","key":"10.3233\/JIFS-223439_ref27","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s10700-020-09342-9","article-title":"Numerical solution and parameter estimation for uncertain SIR model with application to COVID-19","volume":"20","author":"Chen","year":"2021","journal-title":"Fuzzy Optimization and Decision Making"},{"key":"10.3233\/JIFS-223439_ref28","doi-asserted-by":"crossref","unstructured":"Brauer F. , Castillo-Chavez C. and Castillo-Chavez C. , Mathematical models in population biology and epidemiology, Springer, New York, 2nd edition, 2012.","DOI":"10.1007\/978-1-4614-1686-9"},{"key":"10.3233\/JIFS-223439_ref29","unstructured":"Khalil H.K. , Nonlinear systems, Prentice hall, Upper Saddle River, 3nd edition, 2002."},{"key":"10.3233\/JIFS-223439_ref30","doi-asserted-by":"crossref","unstructured":"Kaczorek T. , Selected problems of fractional systems theory (Vol. 411), Springer-Verlag, Berlin, 2011.","DOI":"10.1007\/978-3-642-20502-6"}],"container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"original-title":[],"link":[{"URL":"https:\/\/content.iospress.com\/download?id=10.3233\/JIFS-223439","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T08:51:34Z","timestamp":1769676694000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/full\/10.3233\/JIFS-223439"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,4]]},"references-count":30,"journal-issue":{"issue":"5"},"URL":"https:\/\/doi.org\/10.3233\/jifs-223439","relation":{},"ISSN":["1064-1246","1875-8967"],"issn-type":[{"value":"1064-1246","type":"print"},{"value":"1875-8967","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,11,4]]}}}