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In this paper, we propose a vaccination–age S V I R model with general nonlinear incidence rate and general treatment function. Firstly, we establish the existence, uniqueness and positivity of solutions for the considered S V I R model in the case where the linear part is nondensely defined and satisfies the Hille–Yosida condition by the use of the integrated semigroup theory. Secondly, based on the basic reproduction number R 0 , we investigate the existence and the local stability of the equilibrium points. Moreover, we discuss the global stability of the equilibrium points under the light of the Lyapunov–LaSalle approach. Finally, an application with numerical simulations is done to check the validity of our theoretical results and to exemplify the impact of the treatment and the vaccination on the disease propagation. [ABSTRACT FROM AUTHOR]