Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission

Lyapunov functions for two-dimension SIR and SIRS compartmental epidemic models with non-linear transmission rate of a very general form f(S. I) constrained by a few biologically feasible conditions are constructed. Global properties of these models including these with vertical and horizontal transmission, are thereby established. It is proved that, under the constant population size assumption, the concavity of the function f(S. I) with respect to the number of the infective hosts I ensures the uniqueness and the global stability of the positive endemic equilibrium state.