Developing physiologically meaningful mathematical models that describe multilevel regulation in a complex network of immune processes, in particular, of the system of interferon-regulated virus production processes, is a fundamental scientific problem, within the framework of an interdisciplinary systems approach to research in immunology. Here, we have presented a detailed high-dimensional model describing HIV (human immunodeficiency virus) replication, the response of type I interferon (IFN) to the virus infection of the cell, and suppression of the action of IFN-induced proteins by HIV accessory proteins. As a result, this model includes interactions of all three processes for the first time. The mathematical model is a system of 37 nonlinear ordinary differential equations including 78 parameters. Importantly, the model describes not only the processes of the IFN response of the cell to virus infection, but also the mechanisms used by the virus to prevent effects of the IFN system.