Some infectious disease models with population dynamics and general contact rates

We consider models for the spread of infectious diseases which include nonlinear population dynamics, contact rates which depend on total population size, and variable infective periods. We show that there is a single asymptotically stable equilibrium for diseases with recovery either with no immunity or with full immunity. This equilibrium is the disease-free equilibrium if the contact number is less than one and an endemic equilibrium if the contact number exceeds one.