I am working with the SIR model with vital dynamics:
\begin{align} \frac{dS}{dt} &= \mu (K - S)- \beta SI \label{eq3}\\ \frac{dI}{dt} &= I(\beta S - \gamma -\mu) \label{eq4} \end{align}
where I discarded the last ODE since it does not appear in the first two. The region for this problem is $D={(S,I) : S>0, I \geq 0, S+I \leq K}$
I found that one equilibrium point is $(K,0)$ which is a local asymptotically stable point if $\frac{\beta K}{\gamma + \mu} < 1$. How can I show that is indeed a globally stable point? I thought about using the lyapunov function $V(S,I) = S+I-K$ from which get $\dot{V} = \mu(K-S-I) - \gamma I$. But since $S+I\leq K $ then I cannot conclude that $\dot{V}<0$. What am I doing wrong? Do I need to choose another lyapunv function?
Thank you
I suggest you try another approach, namely that you rule out periodic orbits using the Dulac–Bendixson theorem with $1/I$ as your Dulac function.