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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.

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    $\begingroup$ If it is meant that the balance should hold in the case $\psi(S,I)=\psi S$, then the statement just doesn't hold water: take $\beta=0$. Then the LHS is $I(0)$ and the RHS is $S(0)$, which are different in general. If you insist on some spread, take it small ($\beta\approx 0$) and observe that it can change the values only by $\int_0^\infty \beta SI/N \le \beta \int_0^\infty S$ but $S$ decreases at least exponentially at the rate $\psi$ so that product is much smaller than $S(0)$ when $\beta\to 0$ and it can change nothing. Apparently the author just doesn't know what he/she is talking about... $\endgroup$
    – fedja
    Commented Jan 17, 2023 at 1:20
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    $\begingroup$ Your link is not to the source, but just a screenshot of some page in the source. Please provide the full reference. $\endgroup$
    – YCor
    Commented Feb 16, 2023 at 13:11

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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.

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  • $\begingroup$ Hey Carlo, thanks to you and fedja for you answers. Your equation $1=\int_{0}^{\infty}(αI(t)+ψS(t))dt$ was exactly my solution which is obviously nowhere near the last equation in my source. So I understand that you can't make sense of what the author meant by that either, is that right? $\endgroup$
    – Rick
    Commented Jan 17, 2023 at 20:09

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