Infection-induced host extinction: Deterministic and stochastic models for environmentally transmitted pathogens

Amphibian decline and extinction have been observed on a global scale, highlighting the urgency of identifying the underlying factors. This issue has long been recognized as a critical concern in conservation ecology and continues to receive significant attention. Pathogen infection, in particular the chytrid fungus Batrachochytrium dendrobatidis, is postulated as a key factor contributing to the decline of certain species within specific regions. In this paper, we focus on identifying the pathogen characteristics that can drive host species extinction. Both deterministic and stochastic modeling frameworks based on a susceptible-infectious-pathogen epidemic model are proposed, to assess the influence of pathogen infection on species decline and extinction. Various indices, including the reproduction numbers of the host species, the replication of the pathogen, and the transmission of the pathogen are derived. Theoretical analysis includes the stability of equilibria, the extinction and persistence of host species in the deterministic model, and the evaluation of extinction probability and average extinction time in the stochastic model. Additionally, numerical simulations are conducted to quantify the effects of various factors on host decline and extinction, as well as the probabilities of extinction. We find two crucial conditions for a pathogen to drive host extinction: (i) the pathogen's self-reproduction capacity in the environment, and (ii) the pathogen's impact on the fecundity and survival of the infected host. These findings provide insights that could aid in the design and implementation of effective conservation strategies for amphibians.