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 {Sn:n ∈ N}, the normalization processes {Wˆn:n ∈ N} are studied, and the sufficient conditions of {Wˆn:n ∈ 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 {In:n ∈ N}, the normalization processes {W¯n:n ∈ N} are studied, and the sufficient conditions of {W¯n:n ∈ N} a.s., and L1 convergence are obtained.