SIR Model with Stochastic Transmission

The Susceptible-Infected-Recovered (SIR) model is the cornerstone of epidemiological models. However, this specification depends on two parameters only, which implies a lack of flexibility and the difficulty to replicate the volatile reproduction numbers observed in practice. We extend the classic SIR model by introducing nonlinear stochastic transmission, to get a stochastic SIR model. We derive its exact solution and discuss the condition for herd immunity. The stochastic SIR model corresponds to a population of infinite size. When the population size is finite, there is also sampling uncertainty. We propose a state-space framework under which we analyze the relative magnitudes of the observational and stochastic epidemiological uncertainties during the evolution of the epidemic. We also emphasize the lack of robustness of the notion of herd immunity when the SIR model is time discretized.