Relaxation Oscillation in SEIR Epidemic Models with the Intrinsic Growth Rate

The periodic oscillation transmission of infectious diseases is widespread, deep understanding of this periodic pattern and exploring the generation mechanism, and identifying the specific factors that lead to such periodic outbreaks, which are of very importanceto predict and control the spread of infectious diseases. In this study, to further reveal the mathematical mechanism of spontaneous generation of periodic oscillation solution, we investigate a type of SEIR epidemic model with a small intrinsic growth rate. By utilizing the singular perturbation theory and center manifold theorem, we extend the relaxation oscillation of three-dimensional SIR models to the four-dimensional SEIR models and prove the existence of stable relaxation oscillation with a large amplitude in the model. Numerical simulations are performed to verify our theoretical results. The results presented in this study provide a new idea for the study of the intrinsic mechanism of periodic oscillation in epidemiology, enrich the dynamics of epidemic models, and deepen the understanding of the global dynamics of these models.