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 [Formula: see text] , the normalization processes [Formula: see text] are studied, and the sufficient conditions of [Formula: see text] a.s., [Formula: see text] and [Formula: see text] 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 [Formula: see text] , the normalization processes [Formula: see text] are studied, and the sufficient conditions of [Formula: see text] a.s., and [Formula: see text] convergence are obtained.