{"title":"具有敏感扩散的SIR斑块模型的全局动力学。","authors":"Luju Liu, Weiyun Cai, Yusen Wu","doi":"10.1186/1687-1847-2012-131","DOIUrl":null,"url":null,"abstract":"<p><p>An <math><mi>S</mi> <mi>I</mi> <mi>R</mi></math> epidemiological model with suscptibles dispersal between two patches is addressed and discussed. The basic reproduction numbers <math><msub><mi>R</mi> <mn>01</mn></msub> </math> and <math><msub><mi>R</mi> <mn>02</mn></msub> </math> are defined as the threshold parameters. It shows that if both <math><msub><mi>R</mi> <mn>01</mn></msub> </math> and <math><msub><mi>R</mi> <mn>02</mn></msub> </math> are below unity, the disease-free equilibrium is shown to be globally asymptotically stable by using the comparison principle of the cooperative systems. If <math><msub><mi>R</mi> <mn>01</mn></msub> </math> is above unity and <math><msub><mi>R</mi> <mn>02</mn></msub> </math> is below unity, the disease persists in the first patch provided <math><msubsup><mi>S</mi> <mn>2</mn> <mrow><mn>1</mn> <mo>∗</mo></mrow> </msubsup> <mo><</mo> <msubsup><mi>S</mi> <mn>2</mn> <mrow><mn>2</mn> <mo>∗</mo></mrow> </msubsup> </math> . If <math><msub><mi>R</mi> <mn>02</mn></msub> </math> is above unity, <math><msub><mi>R</mi> <mn>01</mn></msub> </math> is below unity, and <math><msubsup><mi>S</mi> <mn>1</mn> <mrow><mn>2</mn> <mo>∗</mo></mrow> </msubsup> <mo><</mo> <msubsup><mi>S</mi> <mn>1</mn> <mrow><mn>1</mn> <mo>∗</mo></mrow> </msubsup> </math> , the disease persists in the second patch. And if <math><msub><mi>R</mi> <mn>01</mn></msub> </math> and <math><msub><mi>R</mi> <mn>02</mn></msub> </math> are above unity, and further <math><msubsup><mi>S</mi> <mn>2</mn> <mrow><mn>1</mn> <mo>∗</mo></mrow> </msubsup> <mo>></mo> <msubsup><mi>S</mi> <mn>2</mn> <mrow><mn>2</mn> <mo>∗</mo></mrow> </msubsup> </math> and <math><msubsup><mi>S</mi> <mn>1</mn> <mrow><mn>2</mn> <mo>∗</mo></mrow> </msubsup> <mo>></mo> <msubsup><mi>S</mi> <mn>1</mn> <mrow><mn>1</mn> <mo>∗</mo></mrow> </msubsup> </math> are satisfied, the unique endemic equilibrium is globally asymptotically stable by constructing the Lyapunov function. Furthermore, it follows that the susceptibles dispersal in the population does not alter the qualitative behavior of the epidemiological model.</p>","PeriodicalId":53311,"journal":{"name":"Advances in Difference Equations","volume":null,"pages":null},"PeriodicalIF":4.1000,"publicationDate":"2012-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/1687-1847-2012-131","citationCount":"5","resultStr":"{\"title\":\"Global dynamics for an SIR patchy model with susceptibles dispersal.\",\"authors\":\"Luju Liu, Weiyun Cai, Yusen Wu\",\"doi\":\"10.1186/1687-1847-2012-131\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>An <math><mi>S</mi> <mi>I</mi> <mi>R</mi></math> epidemiological model with suscptibles dispersal between two patches is addressed and discussed. The basic reproduction numbers <math><msub><mi>R</mi> <mn>01</mn></msub> </math> and <math><msub><mi>R</mi> <mn>02</mn></msub> </math> are defined as the threshold parameters. It shows that if both <math><msub><mi>R</mi> <mn>01</mn></msub> </math> and <math><msub><mi>R</mi> <mn>02</mn></msub> </math> are below unity, the disease-free equilibrium is shown to be globally asymptotically stable by using the comparison principle of the cooperative systems. If <math><msub><mi>R</mi> <mn>01</mn></msub> </math> is above unity and <math><msub><mi>R</mi> <mn>02</mn></msub> </math> is below unity, the disease persists in the first patch provided <math><msubsup><mi>S</mi> <mn>2</mn> <mrow><mn>1</mn> <mo>∗</mo></mrow> </msubsup> <mo><</mo> <msubsup><mi>S</mi> <mn>2</mn> <mrow><mn>2</mn> <mo>∗</mo></mrow> </msubsup> </math> . If <math><msub><mi>R</mi> <mn>02</mn></msub> </math> is above unity, <math><msub><mi>R</mi> <mn>01</mn></msub> </math> is below unity, and <math><msubsup><mi>S</mi> <mn>1</mn> <mrow><mn>2</mn> <mo>∗</mo></mrow> </msubsup> <mo><</mo> <msubsup><mi>S</mi> <mn>1</mn> <mrow><mn>1</mn> <mo>∗</mo></mrow> </msubsup> </math> , the disease persists in the second patch. And if <math><msub><mi>R</mi> <mn>01</mn></msub> </math> and <math><msub><mi>R</mi> <mn>02</mn></msub> </math> are above unity, and further <math><msubsup><mi>S</mi> <mn>2</mn> <mrow><mn>1</mn> <mo>∗</mo></mrow> </msubsup> <mo>></mo> <msubsup><mi>S</mi> <mn>2</mn> <mrow><mn>2</mn> <mo>∗</mo></mrow> </msubsup> </math> and <math><msubsup><mi>S</mi> <mn>1</mn> <mrow><mn>2</mn> <mo>∗</mo></mrow> </msubsup> <mo>></mo> <msubsup><mi>S</mi> <mn>1</mn> <mrow><mn>1</mn> <mo>∗</mo></mrow> </msubsup> </math> are satisfied, the unique endemic equilibrium is globally asymptotically stable by constructing the Lyapunov function. Furthermore, it follows that the susceptibles dispersal in the population does not alter the qualitative behavior of the epidemiological model.</p>\",\"PeriodicalId\":53311,\"journal\":{\"name\":\"Advances in Difference Equations\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.1000,\"publicationDate\":\"2012-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1186/1687-1847-2012-131\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Difference Equations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1186/1687-1847-2012-131\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2012/8/1 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"Q1\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Difference Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1186/1687-1847-2012-131","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2012/8/1 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"Mathematics","Score":null,"Total":0}
Global dynamics for an SIR patchy model with susceptibles dispersal.
An epidemiological model with suscptibles dispersal between two patches is addressed and discussed. The basic reproduction numbers and are defined as the threshold parameters. It shows that if both and are below unity, the disease-free equilibrium is shown to be globally asymptotically stable by using the comparison principle of the cooperative systems. If is above unity and is below unity, the disease persists in the first patch provided . If is above unity, is below unity, and , the disease persists in the second patch. And if and are above unity, and further and are satisfied, the unique endemic equilibrium is globally asymptotically stable by constructing the Lyapunov function. Furthermore, it follows that the susceptibles dispersal in the population does not alter the qualitative behavior of the epidemiological model.
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.