{"title":"Some properties of branching processes with random control functions and affected by viral infectivity in random environments.","authors":"Min Ren, Guanghui Zhang","doi":"10.1186/s13662-023-03775-3","DOIUrl":null,"url":null,"abstract":"<p><p>In this paper, a model of branching processes with random control functions and affected by viral infectivity in independent and identically distributed random environments is established, and the Markov property of the model and a sufficient condition for the model to be certainly extinct under some conditions are discussed. Then, the limit properties of the model are studied. Under the normalization factor <math><mo>{</mo><msub><mi>S</mi><mi>n</mi></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>}</mo></math>, the normalization processes <math><mo>{</mo><msub><mover><mi>W</mi><mo>ˆ</mo></mover><mi>n</mi></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>}</mo></math> are studied, and the sufficient conditions of <math><mo>{</mo><msub><mover><mi>W</mi><mo>ˆ</mo></mover><mi>n</mi></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>}</mo></math> a.s., <math><msup><mi>L</mi><mn>1</mn></msup></math> and <math><msup><mi>L</mi><mn>2</mn></msup></math> convergence are given; A sufficient condition and a necessary condition for convergence to a nondegenerate at zero random variable are obtained. Under the normalization factor <math><mo>{</mo><msub><mi>I</mi><mi>n</mi></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>}</mo></math>, the normalization processes <math><mo>{</mo><msub><mover><mi>W</mi><mo>¯</mo></mover><mi>n</mi></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>}</mo></math> are studied, and the sufficient conditions of <math><mo>{</mo><msub><mover><mi>W</mi><mo>¯</mo></mover><mi>n</mi></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>}</mo></math> a.s., and <math><msup><mi>L</mi><mn>1</mn></msup></math> convergence are obtained.</p>","PeriodicalId":72091,"journal":{"name":"Advances in continuous and discrete models","volume":null,"pages":null},"PeriodicalIF":2.3000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10184110/pdf/","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in continuous and discrete models","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1186/s13662-023-03775-3","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, a model of branching processes with random control functions and affected by viral infectivity in independent and identically distributed random environments is established, and the Markov property of the model and a sufficient condition for the model to be certainly extinct under some conditions are discussed. Then, the limit properties of the model are studied. Under the normalization factor , the normalization processes are studied, and the sufficient conditions of a.s., and convergence are given; A sufficient condition and a necessary condition for convergence to a nondegenerate at zero random variable are obtained. Under the normalization factor , the normalization processes are studied, and the sufficient conditions of a.s., and convergence are obtained.