Critical Multitype Branching Processes on a Graph and the Model of the HIV Infection Development

We consider the Crump-Mode-Jagers branching process on an oriented graph in an application to modeling the development of HIV-1 infection in a human organism. For all particles of the same global type, located at each of the verteces or arcs of the graph, different types are assigned. Checking the criticality condition and searching for the eigenvectors of an offspring mean matrix in the critical case for the original process are reduced to an offspring mean matrix for some Galton-Watson process. The last has the types of particles corresponding only to the verteces of the graph.