Global stability of a general, scalar-renewal epidemic model

We investigate the global dynamics of a general Kermack-McKendrick-type 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, $\tau$, 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, $R_0$, represents a sharp threshold parameter such that for $R_0\leq 1$, the infection-free equilibrium is globally asymptotically stable; whereas the endemic equilibrium becomes globally asymptotically stable when $R_0 > 1$, i.e. when it exists. This analysis generalizes a number of previous results derived for the global dynamics of epidemic models.