Alessandro Maria Selvitella, Purdue University, Fort Wayne, IN
Stephen J. Guastello, Marquette University, Milwaukee, WI

Abstract: This study introduces a catastrophe-theoretic framework for modeling the multi-wave trajectories of COVID-19 incidence across U.S. states, addressing key shortcomings of traditional epidemic models. Using daily cumulative case counts from all 50 states and the District of Columbia, we fit quintic polynomials and swallowtail catastrophe models to ln-transformed data. The swallowtail model captures abrupt transitions, wave phenomena, and healthcare workload effects often missed by conventional SIR and SEIR approaches. Polynomial fits were excellent overall, with significantly better performance in Republican states, suggesting deeper differences in epidemic regularity or reporting. The swallowtail model maintained high accuracy across most states' cumulative case trajectories, indicating strong generalizability across heterogeneous epidemic wave shapes and reporting practices. Additionally, by correlating nurse shortages and physician rates with model coefficients, we found that workforce sufficiency substantially shapes key nonlinear epidemic features, highlighting healthcare capacity as a critical control variable in pandemic response strategies. Overall, this research demonstrates the adaptability of catastrophe modeling for epidemic time series and emphasizes the value of integrating nonlinear dynamics methods with detailed, interdisciplinary data sources to better describe and interpret pandemic evolution across U.S. states.

Keywords: COVID-19, swallowtail catastrophe, cusp catastrophe, chaos, workload