Exact Results for the Stochastic SIS Epidemic Model in Densely Populated Environments

Abstract. In this study, we investigate the stochastic dynamics of an extended SIS (susceptible-infected-susceptible) 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, the extinction time distribution of the epidemic, and the distribution of the first-passage time at which the number of infections reaches a certain threshold. 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 the bifurcation threshold of the basic reproduction number for the stochastic SIS model is shown to be larger than that for its deterministic counterpart. Finally, we demonstrate that analyzing the first-passage time distribution can offer early warning for interventions and optimize the allocation of emergency beds. Relevance to Life Sciences. Our findings deepen the understanding of disease transmission in densely populated settings such as workplaces, schools, and households, where stochastic effects and simultaneous transmission to multiple individuals cannot be ignored. We present a systematic analysis of the stochastic SIS model in crowded environments and show how the exact results obtained can guide infectious disease control. The eigenvalues of the stochastic model predict the extinction and outbreak rates of the epidemic, while the extinction time distribution characterizes the disappearance of the disease. The emergence of a bimodal distribution of infected individuals indicates that stochastic effects may give rise to two distinct epidemiological outcomes: rapid extinction or a major outbreak. Furthermore, we demonstrate that analyzing the first-passage time distribution provides a quantitative basis for early warning of intervention measures and rational planning of medical resources, including hospital beds and drug inventories. Mathematical Content. We employ a Markov jump process to model stochastic epidemic dynamics. By using complex analysis techniques, including matrix-valued Cauchy’s integral formula and Cauchy’s residue theorem, we analytically solve the master equation for the stochastic SIS model and derive closed-form expressions for both the time-dependent distribution of infected individuals and the first-passage time distribution. The exact time-dependent distribution is then applied to compute the quasi-stationary distribution of infected individuals and the extinction time distribution of the epidemic. In addition, the quasi-potential of the stochastic SIS model is derived using the large deviation theory, and we use it to construct Wentzel–Kramers–Brillouin approximations of the quasi-stationary distribution and the mean extinction time in the limit of large population size. We also combine the time evolution of the probability distribution with threshold theory to perform a stochastic bifurcation analysis.