{"title":"An Immuno-Epidemiological Model: From Individual-Based Approach to Endemic States","authors":"Katarzyna Pichór, Ryszard Rudnicki","doi":"10.1002/mma.70926","DOIUrl":null,"url":null,"abstract":"<div>\n \n <p>We consider immuno-epidemiological SIS models. The models are based on two factors: the dynamics of the immune status and interactions between susceptible and infected individuals. The starting point is an individual-based model describing interactions between individuals. Through an appropriate limit passage with the number of individuals to infinity, we obtain a model acting on measures describing the distribution of immune status in both epidemic groups. We show that the densities of these measures satisfy a system of partial differential equations with a jump operator. This system generates a continuous semigroup of nonlinear operators on the space of densities. The main results are the existence and uniqueness of an endemic state and a method for determining stationary distributions. We also present simulations of the distribution of the number of infected people over time and the stationary distributions.</p>\n </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 16","pages":"17249-17277"},"PeriodicalIF":2.0000,"publicationDate":"2026-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Methods in the Applied Sciences","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mma.70926","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2026/7/31 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We consider immuno-epidemiological SIS models. The models are based on two factors: the dynamics of the immune status and interactions between susceptible and infected individuals. The starting point is an individual-based model describing interactions between individuals. Through an appropriate limit passage with the number of individuals to infinity, we obtain a model acting on measures describing the distribution of immune status in both epidemic groups. We show that the densities of these measures satisfy a system of partial differential equations with a jump operator. This system generates a continuous semigroup of nonlinear operators on the space of densities. The main results are the existence and uniqueness of an endemic state and a method for determining stationary distributions. We also present simulations of the distribution of the number of infected people over time and the stationary distributions.
期刊介绍：
Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome.
Mathematical Methods in the Applied Sciences is an interdisciplinary journal: therefore, all manuscripts must be written to be accessible to a broad scientific but mathematically advanced audience. All papers must contain carefully written introduction and conclusion sections, which should include a clear exposition of the underlying scientific problem, a summary of the mathematical results and the tools used in deriving the results. Furthermore, the scientific importance of the manuscript and its conclusions should be made clear. Papers dealing with numerical processes or which contain only the application of well established methods will not be accepted.
Because of the broad scope of the journal, authors should minimize the use of technical jargon from their subfield in order to increase the accessibility of their paper and appeal to a wider readership. If technical terms are necessary, authors should define them clearly so that the main ideas are understandable also to readers not working in the same subfield.
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