Criticality in stochastic SIR model for infectious diseases based on path-integral approach

The susceptible-infected-removed (SIR) model provides us with a basic scheme for the analysis of the epidemic infectious diseases such as the COVID-19. In this presentation, we focus on the stochastic SIR model which describes the stochastic time-evolutions of the population sizes for the susceptible, infected, and removed individuals. We consider the master equation (Kolmogorov forward equation) for the infection transmission and recovery processes (SI->II and I->R), and transform it into the Hamiltonian formalism with the Fock space a la quantum physics. According to the Doi-Peliti prescription, furthermore, we introduce the path-integral formalism similar to the quantum field theory, and perform the perturbative and non-perturbative calculations for the time-evolution of the susceptible, infected, and removed populations. We find that the critical value Rc of the basic reproduction number, which determines the spreading or the convergence of the infectious diseases, can be modified by the stochastic effects in comparison to the Rc in the conventional deterministic SIR model.