In this work, we investigate the stability of an SIR epidemic model with a generalized nonlinear incidence rate and distributed delay. The model also includes vaccination term and general treatment function, which are the two principal control measurements to reduce the disease burden. Using the Lyapunov functions, we show that the disease-free equilibrium state is globally asymptotically stable if R 0 ≤ 1, where R 0 is the basic reproduction number. On the other hand, the disease-endemic equilibrium is globally asymptotically stable when R 0 > 1. For a specific type of treatment and incidence functions, our analysis shows the success of the vaccination strategy, as well as the treatment depends on the initial size of the susceptible population. Moreover, we discuss, numerically, the behavior of the basic reproduction number with respect to vaccination and treatment parameters. MSC: 34D03; 92D30

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We investigate the global behaviour of a SIRI epidemic model with distributed delay and relapse. From the theory of functional differential equations with delay, we prove that the solution of the system is unique, bounded, and positive, for all time. The basic reproduction number R 0 for the model is computed. By means of the direct Lyapunov method and LaSalle invariance principle, we prove that the disease free equilibrium is globally asymptotically stable when R 0 < 1. Moreover, we show that there is a unique endemic equilibrium, which is globally asymptotically stable, when R 0 > 1.

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