Global stability of a general, scalar-renewal epidemic model

We investigate the global dynamics of a general Kermack-McKendricktype epidemic model formulated in terms of a system of scalar-renewal equations. Specifically, we consider a model for which both the force of infection and the infected removal rates are arbitrary functions of the infection age, τ , and use the direct Lyapunov method to establish the global asymptotic stability of the equilibrium solutions. In particular, we show that the basic reproduction number, R0, represents a sharp threshold parameter such that for R0 ≤ 1, the infection-free equilibrium is globally asymptotically stable; whereas the endemic equilibrium becomes globally asymptotically stable when R0 > 1, i.e. when it exists. This analysis generalizes a number of previous results derived for the global dynamics of epidemic models.