I am currently preparing a presentation on different modifications of the SIR model. In my sources about the use of vaccines, I came across the following model for a specific rate at which the population is vaccinated.
Concurrent Vaccination In contrast to the previous vaccination scenario we now assume that public health steps up only at the onset of an outbreak, $$\begin{aligned} \dot S &= -\beta \frac{S I}{N} - \psi (S, I) \\ \dot I &= \beta \frac{S I}{N} - \alpha I \\ \dot R &= \alpha I + \psi (S, I) \end{aligned} \tag{6.49} $$ There are different outcomes depending on the choice of the function $\psi$. Assume $\psi (S, I) = \psi S I$ with some number $\psi > 0$. Then the number of remaining susceptible $S_{\infty}$ is independent of $\psi$, however the total size of the epidemic is reduced to $$ T = \frac{\beta}{\beta + \psi} \left( 1 - S_{\infty} \right). \tag{6.50} $$ On the other hand, if we assume $\psi (S, I) = \psi S$, then the point $(0,0,1)$ attracts all solutions. At the end we have a balance between infected and susceptible of the form $$ \alpha \int_0^{\infty} I (t) \, {\rm d} t = \psi \int_0^{\infty} S (t) \, {\rm d} t $$
According to the last equation there shall be a balance between infected and susceptible, but I just can't figure out why. I tried integrating $\dot{R}$ over all time but to no avail.