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This work is concerned with the global dynamics of a Susceptible–Vaccinated–Infected–Isolated–Recovered model, denoted by SVIQR, with vaccination-age. Moreover, the considered model contains a relapse term and a general form of the incidence function. This model is formulated to show the effect of sanitary isolation and vaccination on the disease prevalence when the relapse phenomena occurs. First, we show that the model is mathematically well posed by using the integrated semigroup theory. Second, we prove the existence of equilibrium points based on the basic reproduction number ℛ0ω,ξ. Moreover, by developing appropriate Lyapunov functionals and applying the LaSalles invariance principle, we established the global dynamics of the model, especially, we show that if ℛ0ω,ξ ≤ 1 the the disease-free equilibrium is globally asymptotically stable and, if ℛ0ω,ξ > 1 the endemic equilibrium is globally asymptotically stable. Finally, using the finite difference method, numerical simulations are made to reinforce the theoretical findings and to prove the contribution of the vaccination and the isolation on the disease disappearing. [ABSTRACT FROM AUTHOR]

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]