Time-Inhomogeneous Diffusion Process for the SI Epidemic Model

We consider a time-inhomogeneous diffusion process useful to model the evolution of the infected population in the susceptible- infectious epidemic model in a random environment. We assume that there are no removals, no immunes, and no recoveries from infection. The susceptible individuals become infected through contact with infec- tious individuals. A such model is suitable for describing some classes of micro-parasitic infections to which individuals never acquire a long last- ing immunity and over the course of the epidemic everyone eventually becomes infected. We determine the expression of the transition probabil- ity density function and of its conditional moments. Particular attention is dedicated to the first-passage time problem, by deriving closed form results for the first-passage time density through a constant boundary. For the time-homogeneous process, the behavior of the mean and of the variance of the first-passage time is analyzed and some comparisons between the deterministic model and the obtained diffusion process are provided.