Some properties of branching processes with random control functions and affected by viral infectivity in random environments

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 {S_{n}:nin N}, the normalization processes {hat{W}_{n}:nin N} are studied, and the sufficient conditions of {hat{W}_{n}:nin N} a.s., L^{1} and L^{2} 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 {I_{n}:nin N}, the normalization processes {bar{W}_{n}:nin N} are studied, and the sufficient conditions of {bar{W}_{n}:nin N} a.s., and L^{1} convergence are obtained.