Lyapunov Stability of an SIRS epidemic model with varying population: ecological VS physical approaches

We consider a general SIRS epidemiological transmission model for an endemic infection with variable population including a general assumption of density-dependent mortality and vertical transmission. We analyse global stability by the Lyapunov approach, proposing two alternative Lyapunov functions. The first one is based on traditional ecological arguments, while the second one is a 'physical' Lyapunov function based on the energy of an appropriate second order equation. Both functions allow to prove the global stability of the (unique) endemic state under broad - not equivalent - parametric conditions. This suggests that using different Lyapunov functions provide useful complementary information on the parametric regions where global stability prevails.