Let me try and make sense of this "balance between infected and susceptible".

The differential equations are $$\dot{S}=-\beta SI-\psi S,\;\;\dot{I}=\beta SI-\alpha I,\;\;\dot{R}=\alpha I+\psi S.$$ The sum $N=S+I+R=1$ is time independent. The point $(S,I,R)=(0,0,1)$ is a fixed point, let me assume it's attractive.

Then I just integrate the equation for $\dot{R}$ over all times $t>0$, $$1-R(0)=\int_0^\infty \bigl(\alpha I(t)+\psi S(t)\bigr)\,dt.$$ This is not the equation cited in the OP, for reasons which I do not understand.

Let me try and make sense of this "balance between infected and susceptible".

The differential equations are $$\dot{S}=-\beta SI-\psi S,\;\;\dot{I}=\beta SI-\alpha I,\;\;\dot{R}=\alpha I+\psi S.$$ The sum $N=S+I+R=1$ is time independent. The point $(S,I,R)=(0,0,1)$ is a fixed point, let me assume it's attractive.

Then I just integrate the equation for $\dot{R}$ over all times $t>0$, assuming $R(0)=0$ (no recoveries initially), $$1=\int_0^\infty \bigl(\alpha I(t)+\psi S(t)\bigr)\,dt.$$ This is not the equation cited in the OP, for reasons which I do not understand.