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

TL;DR

This paper addresses predicting outbreak severity in stochastic epidemiological models under uncertain contact rates. It compares white Gaussian, Ornstein-Uhlenbeck (OU), and logarithmic OU (logOU) perturbations using Topological Data Analysis, specifically homological bifurcation plots, to characterize the stationary PDFs of the infected fraction in SIS and the removed fraction in SIR. The results show that logOU noise preserves endemic-like PDFs near the deterministic equilibria even at high noise, while white and OU noise drive mass toward eradication for low $R_0$ and shift peaks away from zero for high $R_0$, with SIR exhibits monostable behavior and peak drifts under all noises. The topology-based approach highlights the critical influence of noise type on disease spread predictions and provides a robust, geometry-driven method to detect phenomenological bifurcations across parameter spaces, aiding outbreak management under uncertainty.

Abstract

We investigate predictions of stochastic compartmental models on the severity of disease outbreaks. The models we consider are the Susceptible-Infected-Susceptible (SIS) for bacterial infections, and the Susceptible -Infected-Removed (SIR) for airborne diseases. Stochasticity enters the compartmental models as random fluctuations of the contact rate, to account for uncertainties in the disease spread. We consider three types of noise to model the random fluctuations: the Gaussian white and Ornstein-Uhlenbeck noises, and the logarithmic Ornstein-Uhlenbeck (logOU). The advantages of logOU noise are its positivity and its ability to model the presence of superspreaders. We utilize homological bifurcation plots from Topological Data Analysis to automatically determine the shape of the long-time distributions of the number of infected for the SIS, and removed for the SIR model, over a range of basic reproduction numbers and relative noise intensities. LogOU noise results in distributions that stay close to the endemic deterministic equilibrium even for high noise intensities. For low reproduction rates and increasing intensity, the distribution peak shifts towards zero, that is, disease eradication, for all three noises; for logOU noise the shift is the slowest. Our study underlines the sensitivity of model predictions to the type of noise considered in contact rate.

Figures (20)

• Figure 1: Susceptible-Infected-Susceptible model with parameters $\lambda$ and $\gamma$ for the contact and curing rates respectively.
• Figure 2: Susceptible-Infected-Removed (SIR) model with parameters $\lambda$ and $\gamma$ for the contact and the curing rates respectively.
• Figure 3: Trajectories for contact rate $\lambda$ having $\bar{\lambda} = 1.4 \text{ per month}$ with white, OU and logarithmic OU noise perturbations, followed by the distributions for each---for a relative noise intensity of 1.0. Basic reproduction number $R_0 = 1.4$ (gonorrhea) was taken for these simulations.
• Figure 4: Autocorrelation for contact rate $\lambda$ having $\bar{\lambda} = 1.4 \text{ per month}$ with OU and logarithmic OU noise perturbations---for a relative noise intensity of 1.0. Basic reproduction number $R_0 = 1.4$ (gonorrhea) was taken for these simulations.
• Figure 5: Superlevel cubical persistence of unit-normalized image data shown. Figures correspond to (a) $K^{0.8}$, (b) $K^{0.5}$, (c) $K^{0.4}$ and (d) $K^{0.2}$. Figures adapted from Tanweer2024Tanweer2024b with modifications.