Why if we consider $\liminf_{t\to +\infty} \|(S(t),I(t))-(S^*,0)\| =0$ is not a contradiction of the instability of DFE point?

Question

The explanation of the second paragraph of the image is not clear to me.

Could you explain in more detail the justification mentioned?

Answer 1

You can look at it this way, the stability of DFE (globally in this case) means any solution will converge to it. On the other hand, instability is the negation of it which means there exists at least one solution that will not converge to it (e.g., this means there may still be some solutions that still converge to it). Hence, saying that DFE is unstable does not rule out the convergence of "some" solutions to it, which means it does not imply the persistence of the disease.