{"title":"Mathematical Models for the Large Spread of a Contact-Based Infection: A Statistical Mechanics Approach","authors":"Marzia Bisi, Silvia Lorenzani","doi":"10.1007/s00332-024-10062-2","DOIUrl":null,"url":null,"abstract":"<p>In this work, we derive a system of Boltzmann-type equations to describe the spread of contact-based infections, such as SARS-CoV-2 virus, at the microscopic scale, that is, by modeling the human-to-human mechanisms of transmission. To this end, we consider two populations, characterized by specific distribution functions, made up of individuals without symptoms (population 1) and infected people with symptoms (population 2). The Boltzmann operators model the interactions between individuals within the same population and among different populations with a probability of transition from one to the other due to contagion or, vice versa, to recovery. In addition, the influence of innate and adaptive immune systems is taken into account. Then, starting from the Boltzmann microscopic description we derive a set of evolution equations for the size and mean state of each population considered. Mathematical properties of such macroscopic equations, as equilibria and their stability, are investigated, and some numerical simulations are performed in order to analyze the ability of our model to reproduce the characteristic features of Covid-19 type pandemics.</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":"368 1","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Nonlinear Science","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00332-024-10062-2","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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Abstract
In this work, we derive a system of Boltzmann-type equations to describe the spread of contact-based infections, such as SARS-CoV-2 virus, at the microscopic scale, that is, by modeling the human-to-human mechanisms of transmission. To this end, we consider two populations, characterized by specific distribution functions, made up of individuals without symptoms (population 1) and infected people with symptoms (population 2). The Boltzmann operators model the interactions between individuals within the same population and among different populations with a probability of transition from one to the other due to contagion or, vice versa, to recovery. In addition, the influence of innate and adaptive immune systems is taken into account. Then, starting from the Boltzmann microscopic description we derive a set of evolution equations for the size and mean state of each population considered. Mathematical properties of such macroscopic equations, as equilibria and their stability, are investigated, and some numerical simulations are performed in order to analyze the ability of our model to reproduce the characteristic features of Covid-19 type pandemics.
期刊介绍：
The mission of the Journal of Nonlinear Science is to publish papers that augment the fundamental ways we describe, model, and predict nonlinear phenomena. Papers should make an original contribution to at least one technical area and should in addition illuminate issues beyond that area''s boundaries. Even excellent papers in a narrow field of interest are not appropriate for the journal. Papers can be oriented toward theory, experimentation, algorithms, numerical simulations, or applications as long as the work is creative and sound. Excessively theoretical work in which the application to natural phenomena is not apparent (at least through similar techniques) or in which the development of fundamental methodologies is not present is probably not appropriate. In turn, papers oriented toward experimentation, numerical simulations, or applications must not simply report results without an indication of what a theoretical explanation might be.
All papers should be submitted in English and must meet common standards of usage and grammar. In addition, because ours is a multidisciplinary subject, at minimum the introduction to the paper should be readable to a broad range of scientists and not only to specialists in the subject area. The scientific importance of the paper and its conclusions should be made clear in the introduction-this means that not only should the problem you study be presented, but its historical background, its relevance to science and technology, the specific phenomena it can be used to describe or investigate, and the outstanding open issues related to it should be explained. Failure to achieve this could disqualify the paper.
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