 Small-delay expansion of an epidemic transfer rate

For a smooth delayed-transfer epidemic model, the incidence $rI(t-\tau )S(t-\tau )$ equals $rIS-r\tau (\dot{I}S+I\dot{S})+O({\tau }^2)$. Substituting the zero-delay equations only inside the correction gives $rIS+\tau rIS\{a+r(I-S)\}+O({\tau }^2)$. The correction has opposite signs in the receiving and departing compartments, preserving their transfer balance.