Stability and Hopf bifurcation for age-structured SVIR epidemic model with different compartment ages and two delays effects

An age-structured SVIR epidemic model with different compartment ages and two delays is investigated. The model is transformed into a non-densely defined abstract Cauchy problem. The basic properties, including the positivity and boundedness of solutions, basic reproduction number R0 and the existence of equilibria, are first derived. The linearized system and characteristic equation at an equilibrium from the corresponding abstract Cauchy problem are also obtained. When R0<1, the local stability of the disease-free equilibrium is proved, and when R0>1 and the two delays vanish, the local stability of the endemic equilibrium is proved. When R0>1, and Assumptions 1 and 2 are satisfied, the existence of Hopf bifurcation are established respectively in the general case with two delays as bifurcation parameters and in three special cases with only one delay as the bifurcation parameter. The results show that disease with the incubation period and the infection period of infection age has very complex dynamical behavior at the endemic equilibrium. Finally, numerical examples validate the theoretical results.