Quarantine threshold in the SIR model ($I_{\ast }+S_{\ast }-\frac{a}{r}\log S_{\ast }=I_0+S_0-\frac{a}{r}\log S_0$)

Before isolation, the closed SIR model has first integral $I+S-(a/r)\log S$. If quarantine occurs at the first crossing $I=I_{\ast }$, the corresponding susceptible count $S_{\ast }$ is the root on the outbreak branch of $I_{\ast }+S_{\ast }-(a/r)\log S_{\ast }=I_0+S_0-(a/r)\log S_0$. Immediate prevention of further transmission makes the number infected among the initial susceptible class equal to $S_0-S_{\ast }$.

When exposed animals remain in a not-yet-infective compartment for a fixed incubation period $\tau$, stopping all new exposure does not cancel the pending case transfers. At isolation time $T$, their number is $E(T)={\int }_{T-\tau }^{T}rI(s)S(s)ds=\tau rI(T)S(T)+O({\tau }^2)$ for smooth profiles. Eventual cases therefore include this pipeline in addition to case transfers already completed by $T$. The interpretation depends on the compartment definition: a strictly unexposed susceptible compartment would instead lose these animals at exposure.