  The susceptible-infectious-recovered (SIR) model describes the evolution of
three species of individuals which are subject to an infection and recovery
mechanism. A susceptible $S$ can become infectious with an infection rate
$\beta$ by an infectious $I$- type provided that both are in contact. The $I$-
type may recover with a rate $\gamma$ and from then on stay immune. Due to the
coupling between the different individuals, the model is nonlinear and out of
equilibrium. We adopt a stochastic individual-based description where
individuals are represented by nodes of a graph and contact is defined by the
links of the graph. Mapping the underlying Master equation into a quantum
formulation in terms of spin operators, the hierarchy of evolution equations
can be solved exactly for arbitrary initial conditions on a linear chain. In
case of uncorrelated random initial conditions the exact time evolution for all
three individuals of the SIR model is given analytically. Depending on the
initial conditions and reaction rates $\beta$ and $\gamma$, the $I$-population
may increase initially before decaying to zero. Due to fluctuations, isolated
regions of susceptible individuals evolve and unlike in the standard mean-field
SIR model one observes a finite stationary distribution of the $S$-type even
for large population size. The exact results for the ensemble averaged
population size are compared with simulations for single realizations of the
process and also with standard mean field theory which is expected to be valid
on large fully-connected graphs.
