Susceptible-infected-recovered 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 results in its lack of flexibility and explains its difficulty to replicate the volatile reproduction numbers observed in practice. We extend the standard SIR model to a semiparametric SIR model, by first introducing a functional parameter of transmission, and then making this function stochastic. This leads to a SIR model with stochastic transmission. Our model is particularly tractable. We derive its closed-form solution and use it to compute key indicators, such as the condition (and the threshold) of herd immunity and the timing of the peak. When the population size is finite and the observations are in discrete time, there is also observational uncertainty. We propose a nonlinear state-space framework under which we analyze the relative magnitudes of the observational and intrinsic uncertainties during the evolution of the epidemic. We emphasize the lack of robustness of the notion of herd immunity when the SIR model is time-discretized.