Rate of recoveries in SIR model

Question

The SIR model used to study the dynamics of epidemics is given be the differential equations \begin{align*} \dot S(t) &= -\beta\,I(t)\,S(t) \\ \dot I(t) &= \beta\,I(t)\,S(t) - \gamma\,I(t) \\ \dot R(t) &= \gamma\,I(t) \end{align*} I don't understand the rationale for the last equation. Assuming that the infection lasts for a (given) time $t^*$, the Ansatz $$ \dot R(t) = \beta\,I(t-t^*)\,S(t-t^*) $$ (and correspondingly $ \dot I(t) = \beta\,I(t)\,S(t) - \beta\,I(t-t^*)\,S(t-t^*) $) appears to be more natural, since it ensures $R(t) \approx I(t-t^*)$ at the start of an epidemic. I.e. the number of people recovering at time $t$ corresponds to the number of people having contracted the infection at time $t-t^*$. What am I missing?

Answer 1

This is just a simple chemical model applied to population dynamics. $$ S+I\xrightarrow{β}2I \\ I\xrightarrow{γ}R $$ It has two reactions, whenever $S$ and $I$ meet, new $I$ is produced by conversion from $S$ at rate $β$. I spontaneously converts to $R$ at rate $γ$.

As said, this is a very simple model to demonstrate some principles. More involved models will have more classes. By passing through different classes (one could for instance divide $I$ into $E=$ "exposed" and different stages of infection and healing) you also get some delay effect.