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An infective individual is currently capable of transmitting the modeled infection. This is distinct from a susceptible individual and, when incubation is modeled, from an exposed individual not yet infectious.

Write $S_0=S_{initial}$ and $I_0=I_{initial}$. Until quarantine and while $I,S>0$, division of the equations givesIntegration from the initial state yields the first integralLet $S_{\ast }$ be the susceptible count at the first time $I=I_{\ast }$. It is the root of an equation reached on the decreasing-$S$ outbreak trajectory ofFor a strictly rising threshold crossing this is the larger admissible root of an equation, with $a/r<S_{\ast }<S_0$; if the threshold equals the peak, $S_{\ast }=a/r$. The assumption that quarantine is reached selects the physically relevant branch and excludes a merely hypothetical declining-branch crossing after an earlier one.

Since there is no incubation period here, all removals from $S$ have already become infection cases by quarantine. No further susceptible individuals are infected after isolation. The requested number among the initially susceptible individuals is therefore

The term $rIS$ is a mass-action incidence rate: each infective individual makes potentially transmitting contacts with susceptible individuals, and the total rate is proportional to the product of the two compartment sizes. An infection transfers one animal from $S$ to $I$, accounting for opposite signs in their equations. The per-capita mortality rate of an infective individual is $a$, giving transfer $aI$ from $I$ to $R$.

There are no births, recoveries or additional deaths in this model. Adding the three equations gives $\dot{S}+\dot{I}+\dot{R}=0$, so the total initial biological population is conserved when the dead compartment is included. Before quarantine, $S$ decreases whenever $I,S>0$; after effective isolation no further transmission is permitted.

The delayed influx $rI(t-\tau )S(t-\tau )$ represents animals exposed one incubation period earlier who enter the infective individual compartment now. They die at the instantaneous rate $aI(t)$ once in that compartment. The displayed bookkeeping removes an animal from $S$ at the same delayed time that it enters $I$, preserving $S+I+R$.

For this bookkeeping, $S$ must include the exposed animals which have not yet completed incubation period. If $S$ were instead defined as strictly unexposed susceptible individuals, its loss term would be the current exposure rate and a separate latent compartment would be needed. The calculations here follow the printed delay differential equation. A full initial-value problem for a delay differential equation also needs a history on $[-\tau ,0]$; the following small-delay expansion of an epidemic transfer rate concerns a smooth compatible history and the regular regime assumed by the question.

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.

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 $S_0-S_{\ast }$.

In a compartmental epidemic model, a susceptible individual is one who can acquire the modeled infection. The susceptible compartment counts these individuals. An exposed but not yet infectious individual belongs to a separate latent class in an SEIR model; models that retain such individuals in a not-yet-infective compartment must state that bookkeeping convention explicitly.