Michael T. Meehan, Daniel G. Cocks, Emma S. McBryde. Global stability properties of renewal epidemic models. 03 October 2018. https://arxiv.org/pdf/1707.03489.pdf
We investigate the global dynamics of a general Kermack-McKendricktype epidemic model formulated in terms of a system of renewal equations. Specifically, we consider a renewal 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.