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

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 : n ∈ N } , the normalization processes { W ˆ n : n ∈ N } are studied, and the sufficient conditions of { W ˆ n : n ∈ 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 : n ∈ N } , the normalization processes { W ¯ n : n ∈ N } are studied, and the sufficient conditions of { W ¯ n : n ∈ N } a.s., and L 1 convergence are obtained.