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 infectious individuals. A such model is suitable for describing some classes of micro-parasitic infections to which individuals never acquire a long lasting immunity and over the course of the epidemic everyone eventually becomes infected. We determine the expression of the transition probability 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.