Considering the impact of age in the transmission of infectious diseases, we present an age-structured model with SIRS type and general incidence function. By reformulating the model as an abstract Cauchy problem and applying theorems related with the Hille-Yosida operator, we investigated the dynamic properties, including stability of equilibria and the condition for Hopf bifurcation due to the destabilization of endemic equilibrium. The results reveal that, if the basic reproductive number is less than 1, the infection-free equilibrium is globally stable, without being influenced by the immune age. Conversely, if the basic reproductive number is larger than 1, the endemic equilibrium may be destabilized by the perturbation of the immune age and a Hopf bifurcation can occur. Meanwhile, numerical simulations are conducted to ill-ustrate the theoretical results and to show the influence of the immune age on the dynamical behaviors of the model.