# Rate of recoveries in SIR model

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?

• 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 $S=0$. – David Mar 31 at 23:18
• @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. – user71769 Apr 2 at 9:29
• $S(t-t^*)$ 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. – David Apr 2 at 22:15

## 1 Answer

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.