数学物理学报(英文版) ›› 2009, Vol. 29 ›› Issue (4): 1081-1094.doi: 10.1016/S0252-9602(09)60087-4

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ON THE BASIC REPRODUCTION NUMBER OF GENERAL BRANCHING PROCESSES

蓝国烈,马志明,孙苏勇   

  1. 1.School of Mathematics and information Sciences, Guangzhou University, Guangzhou 510006, China;
    2.Inst. Appl. Math., Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;3.Graduate University of the Chinese Academy of Sciences
  • 收稿日期:2007-03-27 出版日期:2009-07-20 发布日期:2009-07-20
  • 基金资助:

    This work is supported in part by NSFC and 973 Project

ON THE BASIC REPRODUCTION NUMBER OF GENERAL BRANCHING PROCESSES

 LA Guo-Lie, MA Zhi-Meng, SUN Su-Yong   

  1. 1.School of Mathematics and information Sciences, Guangzhou University, Guangzhou 510006, China;
    2.Inst. Appl. Math., Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;3.Graduate University of the Chinese Academy of Sciences
  • Received:2007-03-27 Online:2009-07-20 Published:2009-07-20
  • Supported by:

    This work is supported in part by NSFC and 973 Project

摘要:

Under a very general condition (TNC condition) we show that the spectral radius of the kernel of a general branching process is a threshold parameter and hence plays a role as the basic reproduction number in usual CMJ processes. We discuss also some properties of the extinction probability and the generating operator of general branching processes. As an application in epidemics, in the final section we suggest a generalization
of SIR model which can describe infectious diseases transmission in an inhomogeneous population.

关键词: general branching process, extinction probability, reproduction kernel, spec-tral radius, TNC condition, basic reproduction number, SIR model

Abstract:

Under a very general condition (TNC condition) we show that the spectral radius of the kernel of a general branching process is a threshold parameter and hence plays a role as the basic reproduction number in usual CMJ processes. We discuss also some properties of the extinction probability and the generating operator of general branching processes. As an application in epidemics, in the final section we suggest a generalization
of SIR model which can describe infectious diseases transmission in an inhomogeneous population.

Key words: general branching process, extinction probability, reproduction kernel, spec-tral radius, TNC condition, basic reproduction number, SIR model

中图分类号: 

  • 60J80