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

Abstract

AbstractIn 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 $\{S_{n}:n\in N\}$ { S n : n ∈ N } , the normalization processes $\{\hat{W}_{n}:n\in N\}$ { W ˆ n : n ∈ N } are studied, and the sufficient conditions of $\{\hat{W}_{n}:n\in N\}$ { W ˆ n : n ∈ N } a.s., $L^{1}$ L 1 and $L^{2}$ 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 $\{I_{n}:n\in N\}$ { I n : n ∈ N } , the normalization processes $\{\bar{W}_{n}:n\in N\}$ { W ¯ n : n ∈ N } are studied, and the sufficient conditions of $\{\bar{W}_{n}:n\in N\}$ { W ¯ n : n ∈ N } a.s., and $L^{1}$ L 1 convergence are obtained.