{"title":"Discrete epidemic models with two time scales.","authors":"Rafael Bravo de la Parra, Luis Sanz-Lorenzo","doi":"10.1186/s13662-021-03633-0","DOIUrl":null,"url":null,"abstract":"<p><p>The main aim of the work is to present a general class of two time scales discrete-time epidemic models. In the proposed framework the disease dynamics is considered to act on a slower time scale than a second different process that could represent movements between spatial locations, changes of individual activities or behaviors, or others. To include a sufficiently general disease model, we first build up from first principles a discrete-time susceptible-exposed-infectious-recovered-susceptible (SEIRS) model and characterize the eradication or endemicity of the disease with the help of its basic reproduction number <math><msub><mi>R</mi> <mn>0</mn></msub> </math> . Then, we propose a general full model that includes sequentially the two processes at different time scales and proceed to its analysis through a reduced model. The basic reproduction number <math> <msub><mover><mi>R</mi> <mo>‾</mo></mover> <mn>0</mn></msub> </math> of the reduced system gives a good approximation of <math><msub><mi>R</mi> <mn>0</mn></msub> </math> of the full model since it serves at analyzing its asymptotic behavior. As an illustration of the proposed general framework, it is shown that there exist conditions under which a locally endemic disease, considering isolated patches in a metapopulation, can be eradicated globally by establishing the appropriate movements between patches.</p>","PeriodicalId":53311,"journal":{"name":"Advances in Difference Equations","volume":"2021 1","pages":"478"},"PeriodicalIF":4.1000,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8556850/pdf/","citationCount":"6","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Difference Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1186/s13662-021-03633-0","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2021/10/30 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 6
Abstract
The main aim of the work is to present a general class of two time scales discrete-time epidemic models. In the proposed framework the disease dynamics is considered to act on a slower time scale than a second different process that could represent movements between spatial locations, changes of individual activities or behaviors, or others. To include a sufficiently general disease model, we first build up from first principles a discrete-time susceptible-exposed-infectious-recovered-susceptible (SEIRS) model and characterize the eradication or endemicity of the disease with the help of its basic reproduction number . Then, we propose a general full model that includes sequentially the two processes at different time scales and proceed to its analysis through a reduced model. The basic reproduction number of the reduced system gives a good approximation of of the full model since it serves at analyzing its asymptotic behavior. As an illustration of the proposed general framework, it is shown that there exist conditions under which a locally endemic disease, considering isolated patches in a metapopulation, can be eradicated globally by establishing the appropriate movements between patches.
期刊介绍:
The theory of difference equations, the methods used, and their wide applications have advanced beyond their adolescent stage to occupy a central position in applicable analysis. In fact, in the last 15 years, the proliferation of the subject has been witnessed by hundreds of research articles, several monographs, many international conferences, and numerous special sessions.
The theory of differential and difference equations forms two extreme representations of real world problems. For example, a simple population model when represented as a differential equation shows the good behavior of solutions whereas the corresponding discrete analogue shows the chaotic behavior. The actual behavior of the population is somewhere in between.
The aim of Advances in Difference Equations is to report mainly the new developments in the field of difference equations, and their applications in all fields. We will also consider research articles emphasizing the qualitative behavior of solutions of ordinary, partial, delay, fractional, abstract, stochastic, fuzzy, and set-valued differential equations.
Advances in Difference Equations will accept high-quality articles containing original research results and survey articles of exceptional merit.