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\documentclass[12pt]{minimal}
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				\begin{document}$\{S_{n}:n\in N\}$\end{document}{Sn:n∈N}, the normalization processes\documentclass[12pt]{minimal}
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				\begin{document}$\{\hat{W}_{n}:n\in N\}$\end{document}{Wˆn:n∈N}are studied, and the sufficient conditions of\documentclass[12pt]{minimal}
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				\begin{document}$\{\hat{W}_{n}:n\in N\}$\end{document}{Wˆn:n∈N}a.s.,\documentclass[12pt]{minimal}
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				\begin{document}$L^{1}$\end{document}L1and\documentclass[12pt]{minimal}
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				\begin{document}$L^{2}$\end{document}L2convergence are given; A sufficient condition and a necessary condition for convergence to a nondegenerate at zero random variable are obtained. Under the normalization factor\documentclass[12pt]{minimal}
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				\begin{document}$\{I_{n}:n\in N\}$\end{document}{In:n∈N}, the normalization processes\documentclass[12pt]{minimal}
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				\begin{document}$\{\bar{W}_{n}:n\in N\}$\end{document}{W¯n:n∈N}are studied, and the sufficient conditions of\documentclass[12pt]{minimal}
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				\begin{document}$\{\bar{W}_{n}:n\in N\}$\end{document}{W¯n:n∈N}a.s., and\documentclass[12pt]{minimal}
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				\begin{document}$L^{1}$\end{document}L1convergence are obtained.