Question about a Lyapunov Function

Question

I'm currently reading this paper about a tuberculosis transmission model and in proving the global stability of the endemic equilibrium, they used the following Lyapunov function: $$ V = (S - S^* \ln S) + a_1(E - E^* \ln E) + a_2(I_1 - I_1^*\ln I_1) + a_3(I_2 - I_2^*\ln I_2) $$ where $a_1$, $a_2$, and $a_3$ are constants to be determined later. However, I noticed that if I evaluate V at the equilibrium point $(S^*, E^*, I_1^*, I_2^*)$, it seems that it does not equate to $0$. But I read that evaluating the Lyapunov function at the equilibrium must yield $0$. Am I missing something here? I only have a basic understanding of Lyapunov functions and it is greatly appreciated if you could help me with this. Thanks!

Answer 1

The value of $V$ at the equilibrium is actually irrelevant (you can always add a constant to $V$ to make it zero if you really want). The important thing is that $V$ has a strict minimum there.