Stability and Hopf bifurcation in delayed age-structured SVIR epidemic model with vaccination and incubation

An age-structured SVIR epidemic model with vaccination and incubation in two delays effects is investigated. In the model, one delay expresses the incubation period of disease and another delay expresses the minimum infection age of which the infected individuals possesses the infection ability. For the stringency of analysis, the model is firstly transformed into a non-densely defined abstract Cauchy problem. The nonnegativity and boundedness of solutions, basic reproduction number R0 and the existence of equilibria are established. The characteristic equations of linearized systems at equilibria are calculated from the corresponding abstract Cauchy problem. When R0≤1, the global stability of disease-free equilibrium is proved, and hence the disease will be deracinated. When R0>1, the local stability and existence of Hopf bifurcation at the endemic equilibrium are established. Particularly, two different cases of Hopf bifurcation are discussed in detail. These results show that the transmission of disease with the incubation period and the minimum infection age have very complex dynamical behavior at the endemic equilibrium. Finally, the numerical examples and simulations are presented to verify the theoretical results and further exhibit the existence of stability switch.