Let me give some observations with respect to your article:
Your recollections about classical SIR are possibly too detailed. A short summary of the most important findings might suffice (without extensive proofs like that of Proposition 1.5).
You could cite e.g. a paper like Hethcote's The Mathematics of Infectious Diseases which summarizes many classical results.
Your Hypothesis 1.6 isn't a hypothesis but an assumption.
Your emphasis on the behaviour for $t \rightarrow -\infty$ seems a bit too strong to me. Interesting things happen (only?) for $t \rightarrow +\infty$. I missed a discussion of initial conditions, instead.
With respect to readability I would suggest to use the variable name $s_{\infty}$ instead of $\Gamma$ (which is the classical name of another transcendental function – next to Lambert's $W$) and to stick to the standard symbol $R_0$ instead of $\kappa$.
Personally I would have been interested why $s_{\infty} = W(r\cdot e^{r})/r$ is the unique solution of $x = e^{r(1-x)}$ with $r = -R_0$. (I had to find out that this is a classical result.)
Hope this helps.