= Solution
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.
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