  We discuss the criticality in the stochastic SIR model for infectious
diseases. We adopt the path-integral formalism for the propagation of
infections among susceptible, infectious, and removed individuals, and perform
the perturbative and nonperturbative analyses to evaluate the critical value of
the basic reproduction number ${\cal R}$. In the perturbation theory, we
calculate the mean values and the variances of the number of infectious
individuals near the initial time, and find that the critical value ${\cal
R}_{\text{c}}=1/3$-$2/3$ should be adopted in order to suppress the stochastic
spread of infections sufficiently. In the nonperturbative approach, we derive
the effective potential by integrating out the stochastic fluctuations, and
obtain the effective Euler-Lagrange equations for the time-evolution of the
numbers of susceptible, infectious, and removed individuals. From the
asymptotic behaviors for a long time, we find that the critical value ${\cal
R}_{\text{c}}=2/3$ should be adopted for the sufficient convergence of
infections. We also find that the endemic state can be generated dynamically by
the stochastic fluctuation which is absent in the conventional SIR model. Those
analyses show that the critical value of the basic reproduction number should
be less than one, against the usually known critical value ${\cal
R}_{\text{c}}=1$, when the stochastic fluctuations are taken into account in
the SIR model.
