Exact results for the stochastic SIS epidemic model in densely populated environments

In this study, we investigate the stochastic dynamics of an extended SIS epidemic model in densely populated environments within a Markov jump process framework. We solve the master equation in closed form and obtain exact solutions of the time-dependent distribution of the number of infected individuals, the quasi-stationary distribution of infected individuals, the extinction time distribution of the epidemic, and the mean extinction time. The approximated quasi-stationary distribution and mean extinction time are also derived using the large deviation theory. Interestingly, we find that the first nonzero eigenvalue of the generator matrix of the Markovian model characterizes the extinction rate of the epidemic, while the second nonzero eigenvalue characterizes its outbreak rate. We also examine the stochastic bifurcation for our model based on the time evolution of the probability distribution and find that the system may exhibit two distinct dynamic patterns: (i) the distribution remains unimodal at all times and (ii) the distribution is unimodal at small and large times and is bimodal at intermediate times. The bifurcation threshold of the basic reproduction number in the stochastic SIS model is shown to be large than that in its deterministic counterpart