Uniform persistence in a prey-predator model with a diseased predator

Following the well-extablished mathematical approach to persistence and its developments contained in Rebelo et al. (Discrete Contin Dyn Syst Ser B 19(4):1155-1170. https://doi.org/10.3934/dcdsb.2014.19.1155, 2014) we give a rigorous theoretical explanation to the numerical results obtained in Bate and Hilker (J Theoret Biol 316:1-8. https://doi.org/10.3934/dcdsb.2014.19.1155, 2013) on a prey-predator Rosenzweig-MacArthur model with functional response of Holling type II, resulting in a cyclic system that is locally unstable, equipped with an infectious disease in the predator population. The proof relies on some repelling conditions that can be applied in an iterative way on a suitable decomposition of the boundary. A full stability analysis is developed, showing how the "invasion condition" for the disease is derived. Some in-depth conclusions and possible further investigations are discussed.