Equation in epidemic SIR model with the influence of vaccinations

Question

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.

Answer 1

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.