Catastrophe Theory in SEIRV Epidemic Modeling

Methodology: This study formulates a 5 × 5 SEIRV (Susceptible-Exposed-Infected-Recovered-Vaccinated) nonlinear epidemic matrix model to investigate epidemic dynamics through catastrophe theory. The nonlinear system is reduced near equilibrium points, and the infected population dynamics are transformed into a cubic structure equivalent to the cusp catastrophe model. Analytical methods are employed to derive conditions for multiple equilibria, bifurcation, and sudden epidemic transitions, and a new theorem is established demonstrating that nonlinear epidemic systems with quadratic infection terms and degenerate Jacobian matrices are locally equivalent to cusp catastrophe dynamics. Results: The results indicate that several equilibrium states, bistability, and bifurcating phenomena are present in the epidemic model, while numerical simulations confirm the theoretical predictions and illustrate catastrophic epidemic outbreaks caused by small variations in epidemiological parameters. Discussion: These findings suggest that epidemic systems may undergo abrupt transitions that cannot be fully captured through conventional linear stability analysis, highlighting the presence of hidden instability thresholds and hysteresis effects in nonlinear epidemic behavior. Findings: The integration of catastrophe theory with epidemic matrix modeling offers a novel mathematical perspective on sudden epidemic transitions and mechanisms of instability in infectious disease systems. Conclusion: The study concludes that catastrophe theory provides a powerful mathematical framework for analyzing epidemic instability and sudden outbreak transitions, offering valuable insights for epidemic prediction and the development of more effective disease control strategies.