  The exact analytical solution in closed form of a modified SIR system where
recovered individuals are removed from the population is presented. In this
dynamical system the populations $S(t)$ and $R(t)$ of susceptible and recovered
individuals are found to be generalized logistic functions, while infective
ones $I(t)$ are given by a generalized logistic function times an exponential,
all of them with the same characteristic time. The dynamics of this modified
SIR system is analyzed and the exact computation of some epidemiologically
relevant quantities is performed. The main differences between this modified
SIR model and original SIR one are presented and explained in terms of the
zeroes of their respective conserved quantities. Moreover, it is shown that the
modified SIR model with time-dependent transmission rate can be also solved in
closed form for certain realistic transmission rate functions.
