An SIR model with distributed delay and a general incidence function is studied. Conditions are given under which the system exhibits threshold behaviour: the disease-free equilibrium is globally asymptotically stable if R0 < 1 and globally attracting if R0 = 1; if R0 > 1, then the unique endemic equilibrium is globally asymptotically stable. The global stability proofs use a Lyapunov functional and do not require uniform persistence to be shown a priori. It is shown that the given conditions are satisfied by several common forms of the incidence function.
McCluskey, C. Connell, "Global Stability of an SIR Epidemic Model with Delay and General Nonlinear Incidence" (2010). Mathematics Faculty Publications. 12.