This paper is concerned with a model for the spread of an epidemic in a closed, homogeneously mixed population in which new infections occur at rate $\beta(z)xy/(x + y)$, where x, y and z denote, respectively, the numbers of susceptible, infective and removed individuals. Thus the infection mechanism depends upon the number of removals to date, reflecting behavior change in response to the progress of the epidemic. For a deterministic version of the model, a recurrent solution is obtained when $\beta(z)$ is piecewise constant. Equations for the total size distribution of the stochastic model are derived. Stochastic comparison results are obtained using a coupling method. Strong convergence of a sequence of epidemics to an unusual birth-and-death process is exhibited, and the behavior of the limiting birth-and-death process is considered. An epidemic model featuring sudden behavior change is studied as an example, and a stochastic threshold result analagous to that of Whittle is derived.
"An epidemic model with removal-dependent infection rate." Ann. Appl. Probab. 7 (1) 90 - 109, February 1997. https://doi.org/10.1214/aoap/1034625253