Use of Topological Data Analysis for the Detection of Phenomenological Bifurcations in Stochastic Epidemiological Models.

We study phenomenological (P-)bifurcations in stochastic compartmental epidemiological models under different noise structures. We consider SIS and SIR models with stochastic contact rate modeled using Gaussian white noise, Ornstein-Uhlenbeck (OU) noise, and logarithmic OU (LogOU) noise. To analyze the resulting stationary distributions, we employ homological bifurcation plots from topological data analysis, enabling automated detection of qualitative changes in distribution structure without requiring analytic stationary solutions. Our results show that Gaussian noise can induce secondary peaks corresponding to disease eradication, whereas LogOU noise preserves unimodality across all parameter ranges considered. Additionally, we observe that increasing noise intensity reduces outbreak severity for low reproduction numbers but increases it for high reproduction numbers. These findings demonstrate that topological diagnostics provide a unified framework for analyzing stochastic epidemiological models under both Gaussian and non-Gaussian noise.