Itô vs Stratonovich stochastic SIR models

We prove that the asymptotic behavior of the Stratonovich counterpart of the Itô’s type stochastic SIR model investigated in Tornatore et al. (2005), Ji and Jiang (2012, 2014) is ruled by the same threshold as the deterministic system. In other words, in contrast to the Itô’s model, the intensity of the noise described through the Stratonovich calculus is not relevant for the extinction of the disease. The Stratonovich interpretation of the model is motivated by the parameter perturbation technique, employed on the disease transmission coefficient and used to implement environmental randomness, in combination with the classical Wong–Zakai approximation argument.