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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.

A per-capita mortality rate $a$ means that the death flux from a biological population of size $I$ is $aI$. With constant rate and no other transfers, $\dot{I}=-aI$ gives exponential survival $I(t)=I(0)e^{-at}$. In an epidemic model this can specify mortality from the infective compartment rather than mortality of the entire population.