Using Lyapunov Functions to Construct Lyapunov Functionals for Delay Differential Equations

Given that a Lyapunov function is known for a particular system, we outline an approach for determining terms in the system that can be replaced by similar terms that include delay, without changing the global stability. The approach is based on adding integral terms to the original Lyapunov function so that the new Lyapunov derivative is still negative semidefinite. This gives a Lyapunov functional for the modified system. The delay structures that are permitted include discrete delay, distributed delay, and combinations of the two. Several examples involving compartmental models in virology and epidemiology are included to show the effectiveness of the method, as well as the ease with which it can be applied.