Some of the most important wildlife diseases involve environmental transmission, with disease control attempted via treatments that induce temporary pathogen resistance among hosts. However, theoretical explanations of such circumstances remain few.

A mathematical model is proposed and investigated to analyse the dynamics and treatment of environmentally transmitted sarcoptic mange in a population of bare-nosed wombats. The wombat population is structured into four classes representing stages of infection, in a model that consists of five non-linear differential equations including the unattached mite population. It is shown that four different epidemiological outcomes are possible. These are: (1) extinction of wombats (and mites); (2) mite-free wombat populations; (3) endemic wombats and mites coexisting, with the wombats’ population reduced below the environmental carrying capacity; and (4) a stable limit cycle (sustained oscillating populations) with wombat population far below carrying capacity. Empirical evidence exists for the first two of these outcomes, with the third highly likely to occur in nature, and the fourth plausible at least until wombat populations succumb to Allee effects. These potential outcomes are examined to inform treatment programs for wombat populations. Through this theoretical exploration of a relatively well understood empirical system, this study supports general learning across environmentally transmitted wildlife pathogens, increasing understanding of how pathogen dynamics may cause crashes in some populations and not others.