For many epidemic models written as

\begin{equation}x (t) = f (x(t))\end{equation}

standard analysis includes calculating the basic reproduction number $\mathcal{R}_0$. Often, for $\mathcal{R}_0 > 1$ there is a globally asymptotically stable endemic equilibrium. In the last decade, this has been demonstrated for many ODE models through the use of Lyapunov functions based on the Volterra function

\begin{equation}

g(x) = x − 1 − \log x.

\label{eq1}

\end{equation}

Other models include delay to better describe certain biological processes and can be written as \begin{equation}x (t) = f (x(t), x(t − \tau))\end{equation}or, more generally,\begin{equation}x (t) = f (x_t )\end{equation}where $ x_t : [−\tau, 0] → {R}^n$ for some $\tau > 0$. Recently, Lyapunov functionals (based on the Volterra function $g$) have been used to resolve the global stability for many disease models with delay. In reviewing the many examples, it becomes clear that the Lyapunov functional for the delay equation is very strongly related to the Lyapunov function that works for the corresponding ODE.

In this work, we analyze the connection between the Lyapunov functional that works for Equation (4) and the Lyapunov function that works for the corresponding ODE. Using careful mathematical analysis, we obtain a test that allows one to classify, a priori, the terms that can incorporate delay without affecting the global asymptotic behaviour of the system.