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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?

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  • $\begingroup$ For one it gives a system of delay-differential equations. The SIR model models the recovery time as a distribution allowing for different individuals to recover at different rates. And why the $S(t-t^*)$ term? Why does the size susceptible population affect the recovery of infected people? If everyone has the disease people can still recover even though $S=0$. $\endgroup$ – David Mar 31 at 23:18
  • $\begingroup$ @David Thanks for your comment. The term involving the $S(t-t^*)$ expression reflects the simple idea that all people newly infected at $t$ (viz. $\beta\,I(t)\,S(t)$) will recover at $t+t^*$. I agree that the delay-differential equation is awkward from a mathematical point of view. $\endgroup$ – user71769 Apr 2 at 9:29
  • $\begingroup$ $S(t-t^*)$ has no connection to number of newly infected people. If you want everyone to have the disease for exactly $t^*$ days, then write $R(t)=I(t-t^*)$, no differential equation necessary. $\endgroup$ – David Apr 2 at 22:15
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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.

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