A stochastic susceptible-infected-susceptible epidemic model with Stratonovich processes

We investigate the susceptible-infected-susceptible (SIS) epidemic model under fluctuating random environments. We consider the effect of uncertainty in estimation of parameters in the SIS model, the infection rate and the recovery rate, by using the technique of parameter perturbation. Previous works on parameter perturbation employed the Itô stochastic differential equation (SDE) to examine the effect of environmental stochasticity and found that epidemic thresholds occur at the reduced value of the basic reproduction number by the amount proportional to the variance of the perturbation. We employ the Stratonovich SDE, which seems to be more applicable for parameter perturbation in the epidemic model as well as the Itô SDE and find that the effect of environmental stochasticity in the Stratonovich SDE vanishes when the fluctuation is Gaussian, whereas the effect in the Itô SDE is proportional to the variance. Furthermore we carry out numerical simulations in the form of the Gillespie algorithm and find that the results of simulations are consistent with the analytic predictions.