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 {Snn∈N}, the normalization processes {Wˆnn∈N} are studied, and the sufficient conditions of {Wˆnn∈N} a.s., L1 and L2 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 {Inn∈N}, the normalization processes {W¯nn∈N} are studied, and the sufficient conditions of {W¯nn∈N} a.s., and L1 convergence are obtained.