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 $\{S_n:n\in N\}$, the normalization processes $\{{\hat{W}}_n:n\in N\}$ are studied, and the sufficient conditions of $\{{\hat{W}}_n:n\in N\}$ 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 $\{I_n:n\in N\}$, the normalization processes $\{{\overline{W}}_n:n\in N\}$ are studied, and the sufficient conditions of $\{{\overline{W}}_n:n\in N\}$ a.s., and $L^1$ convergence are obtained.