In this paper we introduce a new technical development of branching processes that can potentially model spread of epidemics by including various stages of infection. In fact, we present an epidemic model that the disease spreads from an individual to another one in such a way that the ability of individual u to infect a susceptible individual depends only on how long u has been infected. For analysis of this model, we introduce a new and special model of the Crump-Mode-Jagers branching processes in discrete time and obtain some of its fundamental properties.
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