I’m trying to find a suitable lyapunov function for my paper and come accross this paper about SIR model where they used $V(x,y)= (S-S^1)^2 + (I-I^1)^2$ where $S^1$ and $I^1$ are endemic equilibriums. Here’s the link to the paper:
https://www.math.lsu.edu/system/files/Project%202%20Final%20Paper.pdf
Now, based from the paper given that $V(x,y)= (S-S^1)^2 + (I-I^1)^2$,
\begin{equation} \tag{1} dV(x,y)= 2(S-S^1)S’ + 2(I-I^1)I’ \end{equation}
\begin{equation} \tag{2} dV(x,y)= -\frac{2}{S^1}(S-S^1)^2 - \frac{2}{I^1}(I-I^1)^2 \end{equation}
With that, I’m very confused on how the transformation from equation # 1 to equation #2 happened. Can someone help me with this?? Thanks a alot in advance
Here’s the SIR equation used in the paper: \begin{equation} S’= \Lambda -\frac{\beta S}{N}-\mu S \end{equation} \begin{equation} I’= \frac{\beta S}{N}-(\mu + \gamma)I \end{equation} \begin{equation} R’= (\gamma + \mu)I - \mu (N-S) \end{equation}
Where
\begin{equation} N= S+I+R \end{equation}
Also, \begin{equation} S^1= \frac{\Lambda ^2}{\mu (\beta + \Lambda)} \end{equation} \begin{equation} I^1= \frac{\Lambda \beta}{ (\beta + \Lambda) + (\mu + \gamma)} \end{equation}