 Solution

The law of mass action assigns infection rate $rIS$, death rate $aI$, and vaccination rate $pIS$. Vaccination consumes a susceptible animal and produces a vaccinated one while leaving the triggering infected animal present. In dimensional time $T$,All compartment counts remain nonnegative. Adding the equations gives $(I+S+V+R{)}_T=0$; deaths are retained in the bookkeeping compartment rather than removed from that conserved count. There is no recovery class of live immune survivors in the infection reaction: in this particular model every animal that becomes infected ultimately dies.